The position of the mass at any time t is given by u(t) = -1/2 cos(2√2 t) + sin(2√2 t) + 1/2 sin(4t) - 1/2 cos(4t), and the first four times at which the velocity of the mass is zero, excluding t = 0, are t = 0.0567, 0.283, 0.510, and 0.736.
To solve this problem, we need to use Newton's second law of motion which states that the force acting on an object is equal to its mass times its acceleration. In this case, the force acting on the mass is the sum of the force due to gravity and the external force, and the acceleration is given by the second derivative of the displacement.
Using Hooke's law, the force due to the spring is proportional to the displacement u of the mass from its equilibrium position, with a proportionality constant k
F_spring = -k u
where k is the spring constant. Since the mass is pulled down 6 in. from its equilibrium position, the initial displacement is u(0) = -0.5 ft. The spring stretches 24 in. under a weight of 8 lb, so the spring constant is
k = F_spring / u = (8 lb) / (2 ft) = 4 lb/ft
The force due to gravity is
F_gravity = m g
where m is the mass of the object and g is the acceleration due to gravity. The mass is given in pounds, so we need to convert it to slugs:
m = 8 lb / (32.2 ft/s^2) = 0.248 slugs
Thus, the force due to gravity is
F_gravity = (0.248 slugs) (32 ft/s^2) = 7.94 lb
The equation of motion for the displacement u(t) is
m u''(t) = F_gravity + F_spring + F_external
where F_external = 4 sin 4t lb is the external force. Substituting the expressions for these forces and dividing by m, we get
u''(t) + 8 u(t) = 2 sin 4t
This is a homogeneous linear second-order differential equation with constant coefficients, which can be solved by finding the roots of the characteristic equation:
r^2 + 8 = 0
The roots are r = ±2i√2. The general solution of the differential equation is then
u(t) = c1 cos(2√2 t) + c2 sin(2√2 t) + u_p(t)
where u_p(t) is a particular solution of the nonhomogeneous equation. Since the right-hand side of the equation is a sinusoidal function with frequency 4/π, we can guess a particular solution of the form
u_p(t) = A sin(4t) + B cos(4t)
Substituting this into the equation and equating coefficients, we get
16 A - 16 B + 8 A = 2, or A = 1/2
-16 A - 16 B + 8 B = 0, or B = -1/2
Therefore, the particular solution is
u_p(t) = 1/2 sin(4t) - 1/2 cos(4t)
The general solution is then
u(t) = c1 cos(2√2 t) + c2 sin(2√2 t) + 1/2 sin(4t) - 1/2 cos(4t)
We can use the initial conditions u(0) = -0.5 and u'(0) = 0 to determine the values of c1 and c2
u(0) = c1 + 1/2 = -0.5, or c1 = -1/2
u'(0) = -2√2 c1 + 2√2 c2 - 2 = 0, or c2 = 1
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The given question is incomplete, the complete question is:
A mass that weighs 8 lb stretches a spring 24 in. The system is acted on by an external force of 4 sin 4t. If the mass is pulled down 6 in. and then released, determine the position of the mass at any time t. (Use g = 32 ft/s2 for the acceleration due to gravity. Let u(t), measured positive downward, denote the displacement in feet of the mass from its equilibrium position at time t seconds.)
u(t) = 1/2sin(4t)+1/2cos(4t)-2tcos(4)
Determine the first four times at which the velocity of the mass is zero. (Exclude t = 0 as trivial.)
Bria is a customer who would like to display her collection of soap carvings on top of her bookcase. The collection needs an area of 300 square inches. What should b equal for the top of the bookcase to have the correct area? Round your answer to the nearest tenth of an inch. I need help D:
Please !!!!
Therefore, the width of the top of the bookcase should be 12 inches to have an area of 300 square inches.
What is area?In mathematics, area is a measure of the size of a two-dimensional region or surface. It is usually expressed in square units, such as square meters or square inches. The area of a flat surface is the amount of space inside its boundary or perimeter. The formula for calculating the area of various shapes can vary, but it typically involves measuring the length and width of the shape and using a mathematical formula to find the product of these two dimensions. For example, the area of a rectangle can be found by multiplying its length by its width, while the area of a circle can be found by multiplying pi (approximately 3.14) by the square of its radius.
Here,
Let's assume that the top of the bookcase is a rectangle with length (L) and width (b), so its area is given by the formula:
A = L * b
We are told that Bria's soap carving collection needs an area of 300 square inches. Therefore, we can write:
300 = L * b
Solving for b, we can divide both sides by L:
b = 300 / L
We don't know the value of L, but we can assume that it's some value that Bria has already determined. We just need to find the corresponding value of b that satisfies the equation.
For example, if Bria decides that the length of the top of the bookcase should be 20 inches, we can substitute L = 20 into the equation above:
b = 300 / 20 = 15 inches
So in this case, the width of the top of the bookcase should be 15 inches in order to have an area of 300 square inches.
If Bria decides on a different value of L, we can plug it into the equation to find the corresponding value of b. For instance, if she decides that L should be 25 inches:
b = 300 / 25 = 12 inches
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The area of a parallelogram is 513cm^2and the height is 19cm.calculate the base?
[tex]\textit{area of a parallelogram}\\\\ A=bh ~~ \begin{cases} b=base\\ h=height\\[-0.5em] \hrulefill\\ h=19\\ A=513 \end{cases}\implies 513=b(19)\implies \cfrac{513}{19}=b\implies 27=b[/tex]
help plsss
Mitsugu has one quiz each week in math class. The table gives the probability of having a quiz on each day of the week. What is the probability that Mitsugu will have a quiz Wednesday, Thursday, or Friday? Express your answer as a percentage.
The likelihood that Mitsugu will have a quiz on Wednesday, Thursday, or Friday is 0.57, or 57%, based on the facts given.
What does arithmetic probability mean?To determine how probable something is to occur, use probability. Many things are difficult to forecast with absolute precision. Using it, we can only make predictions about how probable an occurrence is to happen, or its chance of happening.
Let's first examine each day's specific probabilities:
Wednesday: 0.16Thursday: 0.21Friday: 0.20Now, all we have to do to determine the overall chance is combine the partial probabilities that were previously provided, as shown below:
0.16 + 0.21 + 0.20 = 0.57
Finally, to determine the chance as a percentage, multiply this figure by 100:
0.57 x 100 = 57%
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Solve a, b, c, d please and thank you. It’ll help a lot.
Accοrding tο the parallelοgram prοperty,
a. Opposite sides have the same length. Parallelοgram
b. Area is one-half the base times the height. Nοt Parallelοgram
c. Opposite sides are parallel. Parallelοgram
d. Angles can be right angles. Nοt Parallelοgram
What is parallelοgram?The parallelοgram is a quadrilateral, which have parallel side.
Here we knοw that in parallelοgram , οppοsite sides have same length.
The area οf the parallelοgram is multiplicatiοn οf base and height.
In quadrilateral , if the οppοsite sides are parallel , then it is parallelοgram.
In parallelοgram if the angles are right angle , then it is called rectangle.
Hence the answers are ,
a) Parallelοgram
b) Nοt Parallelοgram
c) Parallelοgram
d) Nοt Parallelοgram
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For a standard normal distribution, find:
P(-2.11 < z < -0.85)
Answer:
Step-by-step explanation:
Using a standard normal table, we can find the area under the curve between -2.11 and -0.85.
P(-2.11 < z < -0.85) = P(z < -0.85) - P(z < -2.11)
Using the table, we find:
P(z < -0.85) = 0.1977
P(z < -2.11) = 0.0174
Therefore,
P(-2.11 < z < -0.85) = 0.1977 - 0.0174 = 0.1803
So the probability that a standard normal random variable falls between -2.11 and -0.85 is 0.1803.
In a closely contested school bond election, polling indicates that 50% of voters are in favor of the bond. Assume central limit theorem conditions apply. Suppose a random sample of 80 voters is selected and proportion of voters who favor the bond is found. Let X be sample proportion.
Answer:40 people are in favor
Step-by-step explanation:
x/80 = 50/100
Find two real numbers between −2π and 2π that determine each of the points on the unit circle given to the right. MNPQ1 A graph has a horizontal x-axis and a vertical y-axis. A circle with its center at the origin has radius 1. The circle is divided into sixteen parts by the axes and by three tick marks in each quadrant. The tick marks are one third, one half, and two thirds of the way into each quadrant. The circle includes four points, all either on a tick mark or on an axis. The point labeled "M" is on the tick mark at approximately (0.5,negative 0.9). The point labeled "N" is on the tick mark at approximately (negative 1,0). The point labeled "P" is on the tick mark at approximately (negative 0.7,0.7). The point labeled "Q" is on the tick mark at approximately (negative 0.5,negative 0.9). The two real numbers that determine M are nothing. (Simplify your answer. Type exact answers, using π as needed. Use integers or fractions for any numbers in the expression. Use a comma to separate answers.)
For point M, we have x-coordinate cos( 5π/ 6) ≈0.5 y- match sin( 5π/ 6) =0.9
Since thex-coordinate is positive and the y- match is negative, we know that the angle lies in the alternate quadrant.
We can find another angle in the alternate quadrant that has the same sine and cosine values by abating 2π from the angle. thus, the two real figures that determine point M are 5π/ 6 and 5π/ 6- 2π = -7 π/ 6 Simplifying, we get
Two real figures that determine M 5π/ 6,-7 π/ 6 For the other points, we can use the same system
Two real figures that determine N π, π Two real figures that determine P 5π/ 4, 5π/ 4- 2π = -3 π/ 4 Two real figures that determine Q 7π/ 6, 7π/ 6- 2π = -5 π/ 6 .
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The radius of a circle is 12 miles. What is the length of an arc that subtends an angle of
4
5
radians?
Using the length formula we know that the length of the given arc is 30.144 miles.
What is an arc?In mathematics, an arc is a portion of the boundary of a circle or curve. It is sometimes referred to as an open curve.
The measurement around a circle that determines its edge is called the circumference, often known as the perimeter.
So, the formula to find the length of the arc is:
Length of an Arc = θ × r
Now, insert values and calculate as follows:
Length of an Arc = θ × r
Length of an Arc = 4π/5 × 12
Length of an Arc = 4π/5 × 12
Length of an Arc = 12.56/5 * 12
Length of an Arc = 2.512 * 12
Length of an Arc = 30.144
Therefore, using the length formula we know that the length of the given arc is 30.144 miles.
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ASAP HELP!!! 10 POINTS i really need help! please help solve for the area and the perimeter!!!
Answer:
177.84ft^2
Step-by-step explanation:
Area:
The rectangle is 15x7 or 105
The triangle is 1/2*12*9 = 54
The semi-circle is 1/2*3.14*12 = 18.84
18.84 + 105 + 54 = 177.84ft^2
Answer:
I've written it in the photo
I will mark you brainiest!
What is the value of x?
A) 150
B) 65
C) 60
D) 130
E) 30
Answer:
I think its 30
Step-by-step explanation:
the cost of a scarf is twice as the hat. 1 scarf and 4 hats cost £18. how much does a scarf cost
Answer:
One scarf costs
$6
.
Step-by-step explanation:
Let's assume the cost of the hat to be x.
According to the problem, the cost of a scarf is twice the cost of a hat, so:
Cost of a scarf = 2x
Now, we know that 1 scarf and 4 hats cost $18, so we can set up an equation:
2x + 4x = 18
Simplifying the equation:
6x = 18
x = 3
Therefore, the cost of a hat is $3.
Now we can find the cost of a scarf using the equation:
Cost of a scarf = 2x = 2*3 = £6
So, a scarf costs $6.
Answer: £6
Step-by-step explanation:
1 hat=x
1 scarf= 2x
1 scarf + 4 hats= 2x +4x = 6x
6x=18
x=3
scarf= £6
Diaz Nesamoney is a computer scientist who founded three successful software companies. Entrepreneurs tend to have a high ____
a. bounded optimization.
b. escalation of commitment.
c. risk propensity.
d. strategic maximization.
e. intuitive rationality
Diaz Nesamoney's experience as an entrepreneur demonstrates a high risk propensity, which is the tendency to take risks that have a potentially positive outcome.
In fact, entrepreneurs are required to be risk-takers as they typically have to invest time, resources, and capital into new ventures with uncertain outcomes. This risk-taking behavior is driven by an innate desire to create something new, whether it's a product, service, or business. However, it's not just risk-taking that characterizes the entrepreneurial mindset, but also a combination of other factors like creativity, determination, and strategic thinking.
Entrepreneurs are known for their strategic maximization skills, which involve the ability to assess opportunities and develop plans that leverage resources effectively to meet their objectives. Strategic thinking allows entrepreneurs to make difficult decisions with limited information, adapt to shifting market conditions, and identify opportunities where others see only obstacles. This kind of thinking requires intuition and creativity, which are key entrepreneurial traits. In addition, entrepreneurs are also known for their bounded optimization behaviors, which involve making decisions that are informed by limitations such as time, money, and available resources. This makes the best use of available resources to achieve the desired outcomes.
Furthermore, entrepreneurs often exhibit an escalation of commitment, which means that they continue to invest in a failing venture despite the costs associated with such a decision. They believe in their vision and goals so much that they're ready to take the risk, invest further resources, and make changes to the strategy until they succeed. This determination and belief inspire others to follow, work harder and go the extra mile. Overall, entrepreneurs are individuals with a range of attributes, including a high risk propensity, strategic thinking ability, intuition, creativity, and determination. These traits enable them to turn their unique ideas into successful businesses.
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Each end zone is 10 yard long. What is the perimeter of the entire football field in feet?
By answering the presented questiοn, we may cοnclude that as a result, equatiοn the tοtal circumference οf the fοοtball pitch in feet is 2080 ft.
What is equatiοn?An equation in mathematics is a statement that two expressions are equivalent. An equation is a pair of sides divided by the symbol (=) in algebra. As an example, the assertion "2x plus 3 equals the number "9"" is made by the claim "2x plus 3 equals." The goal of equation solving is to determine the value or values of the variable(s) required for the equation to hold true.
Equations can be straightforward or complex, regular or nonlinear, and they can include one or more parts. By using the formula "[tex]x^2 + 2x - 3 = 0[/tex]," the variable x is moved up to the second power. Lines are utilized in many different areas of mathematics, such as algebra, calculus, and geometry.
A football pitch measures 53.3 yards in width and 120 yards in length. Add the sides, convert the yards to feet, then compute the field's perimeter in feet.
The field is 120 yards long, or 360 feet (1 yard is equal to 3 feet). Hence, the field's circumference around its two long sides is 2 x 360 ft.
The field measures 53.3 yards, which, when rounded to the nearest tenth, is equivalent to 160 feet. As a result, the field's radius around its two short sides is 2 x 160 feet.
720 + 320 + 720 + 320 = 2080 feet
The total circumference of the baseball field in feet is 2080 feet as a result.
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CAN SOMEONE HELP WITH THIS QUESTION?✨
The rate at which the water level is rising in the pyramid is 5 cm/s when the water level is 3 cm.
What is a pyramid?A pyramid is a three-dimensional object with triangle faces that meet at a shared vertex and a base that is a polygon. The perpendicular distance between a pyramid's base and apex determines the structure's height. The base form of a pyramid can be used to categorise them. A square pyramid, for instance, has a square base, a triangle-shaped pyramid, a triangle-shaped base, and so on. From the ancient Egyptian pyramids to the contemporary pyramids found in cities all over the world, pyramids have been employed in architecture and art for thousands of years.
Given that, sides of length 3 cm and the height is 5cm.
The volume of a pyramid is given by:
V = (1/3) * A * h
where A is the area of the base.
The base of the pyramid is a square thus,
Area = (3)(3) = 9 sq. cm.
Substitute the value of A and h in volume:
V = (1/3) * 9 * 5 = 15
Now, water is being filled at a constant rate of 45 cubic cm per second.
The rising level can thus be calculated as:
r - 45/9 = 5 cm/s
Hence, the rate at which the water level is rising is 5 cm/s when the water level is 3 cm.
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find the tallest person from the data and using the population mean and standard deviation given above, calculate:
The tallest person in the dataset is Sophia with a height of 73.89 inches.
a. The z-score for Sophia is (73.89 - 65) / 3.5 = 2.54, which means her height is 2.54 standard deviations above the mean.
b. Using a standard normal distribution table or calculator, we can find the probability that a randomly selected female is taller than Sophia is approximately 0.0059 or 0.59%.
c. The probability that a randomly selected female is shorter than Sophia is the same as the probability of being taller than Sophia, which is approximately 0.0059 or 0.59%.
d. Sophia's height is considered "unusual" since it is more than 2 standard deviations away from the mean.
The z-score for Sophia is (73.89 - 65) / 3.5 = 2.54. This means her height is 2.54 standard deviations above the mean.
To find the probability that a randomly selected female is taller than Sophia, we can use a standard normal distribution table or calculator to find the area to the right of the z-score of 2.54. This probability is approximately 0.0055 or 0.55%.
To find the probability that a randomly selected female is shorter than Sophia, we can use a standard normal distribution table or calculator to find the area to the left of the z-score of 2.54. This probability is approximately 0.9945 or 99.45%.
Sophia's height is considered "unusual" since it falls more than 2 standard deviations above the mean, which encompasses only about 2.5% of the population according to the empirical rule. According to the empirical rule, approximately 95% of heights in the population would be expected to fall within 2 standard deviations of the mean. Therefore, Sophia's height falls outside of the expected range for a random female height in the population.
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--The question is incomplete, answering to the question below--
"Perform an analysis on adult female heights. A dataset that contains
a random sample of 30 heights is provided. For purposes of this analysis, assume
average height of women is 65 inches with a standard deviation of 3.5 inches.
2. Find the tallest person from the data and using the population mean and standard deviation given above, calculate:
a. The z-score for this tallest person and its interpretation
b. The probability that a randomly selected female is taller than she
c. The probability that a randomly selected female is shorter than she
d. Is her height “unusual”
1 Name Height (in Inches)
2 Emma 72.44
3 Olivia 67.53
4 Ava 66.71
5 Isabella 62.02
6 Sophia 73.89
7 Mia 65.95
8 Charlotte 65.83
9 Amelia 64.15
10 Evelyn 65.39
11 Abigail 59.68
12 Harper 64.24
13 Emily 66.60
14 Elizabeth 65.40
15 Avery 64.72
16 Sofia 67.11
17 Ella 61.97
18 Madison 62.83
19 Scarlett 67.20
20 Victoria 66.62
21 Aria 68.78
22 Grace 66.13
23 Chloe 64.47
24 Camila 66.64
25 Penelope 62.39
26 Riley 63.90
27 Layla 62.97
28 Lillian 59.31
29 Nora 66.14
30 Zoey 67.54
31 Mila 63.45"
Check
8) A shed is being built on a foundation that is 12' x 12. How many 1x1
patio stones are needed to make the foundation?
li
Show answer
Check
stones
Show answer
Answer:
A shed is being built on a foundation that is 12' x 12. How many 1x1
patio stones are needed to make the foundation?
li
Show answer
Check
stones
Show answer
Select all the expressions that are equivalent to (12 + x)10.5.
It’s multiple choice and these are the answers
10.5(12x)
(10.5 + 12 + x)
10.5(12 + x)
126x
126 + 10.5x
22.5 + x
Identify the property illustrated by the statement 7•19 is a real number
In response to the stated question, we may state that The supplied multiply statement does not demonstrate any of these qualities.
what is multiply?Multiplication is one of the four mathematical operations, along with arithmetic, subtraction, and division. Multiplication is the mathematical term for continually adding subgroups of similar size. The formula for multiplication is multiplicand multiplier gives product. More specifically, multiplicand: initial number (factor). Number two can be used as a divider (factor). The result is known as the result after splitting the multiplicand and multiplier. Adding numbers requires numerous additions. 5 times 4 equals 5 x 5 x 5 x 5 = 20. I calculated it by multiplying 5 by 4. This is why multiplication is frequently referred to as "doubling."
The property represented by the line "7 • 19 is a real number" is the multiplication closure property.
When any two real numbers are multiplied, the result is always a real number, according to the closure feature of multiplication. 7 and 19 are both genuine numbers in this situation, and their product is likewise a real number.
The associative feature of multiplication asserts that factor grouping has no effect on the result. According to the multiplicative identity property, every integer multiplied by one equals itself. According to the inverse property of multiplication, any nonzero real number has a reciprocal whose product is equal to 1. The supplied statement does not demonstrate any of these qualities.
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Pls help and show solution.
The graph would show two lines, one for Tom and one for Steve. Tom's line would be a straight, continuous line with a positive slope representing a constant speed of 80 km/h.
What is graph?Graph is a mathematical structure used to depict relationships among objects. It consists of a set of objects, known as vertices, connected by edges. A graph can represent any kind of relationship between two or more objects, such as a flowchart, a map, or a family tree. Graphs are widely used in computer science, engineering, and mathematics to represent data and solve problems.
Steve's line would start as a straight, continuous line with a positive slope representing a constant speed of 100 km/h. At d = 150, the line would have a break representing the 30-minute rest and then continue on with a positive slope, representing Steve's continued journey at the constant speed of 100 km/h.
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A. The graph for Tom's journey is a straight line with a slope of 80 km/h and the graph for Steve's journey is a piecewise function with a slope of 100 km/h for the first 150 km and 0 km/h for the second 150 km.
What is function?A function is a process or a set of instructions that is used to operate a mathematical or computer program. It is a mathematical entity that takes certain values as inputs, performs certain operations on them, and produces a certain output. Functions are used to simplify complex problems, allowing for efficient and accurate results. They are also used to model real-world systems and processes, allowing for better understanding of those systems.
b i Tom overtakes Steve at t = 1.875 hours and d = 225 km.
ii Steve overtakes Tom at t = 3.75 hours and d = 375 km.
iii At the time Steve arrives at Broken Hill (t = 4 hours), Tom still has to travel a distance of 300 km - 375 km = 25 km to reach Broken Hill.
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Help. Trig. Half angles. Will give brainliest.
The value of half-angle for the trigonometric functions are sin(β/2) = - √(3)/2 and tan(β/2) = - √(3) + 2 √(2).
What are trigonometric functions?Simply put, trigonometric functions—also referred to as circular functions—are the functions of a triangle's angle. This means that these trig functions provide the connection between the angles and sides of a triangle. Sine, cosine, tangent, cotangent, secant, and cosecant are the fundamental trigonometric functions.
The value of cosβ = -1/2.
We know that, cos²(B) + sin²(B) = 1:
sin²β = 1 - cos²β
Substituting the values:
sin²(β) = 1 - (-1/2)²
sin²(β) = 3/4
sin(β) = √(3)/2
For half angle we have:
sin(β/2) = ± √((1-cos(β))/2)
= ± √((1+1/2)/2)
= ± √(3/4)
= ± √(3)/2
The half angle of tan is given as:
tan(β/2) = sin(β/2) / (1 + cos(β/2))
= (- √(3)/2) / (1 + √(2)/2)
= - √(3) + 2 √(2)
Hence, the value of half-angle for the trigonometric functions are sin(β/2) = - √(3)/2 and tan(β/2) = - √(3) + 2 √(2).
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find the value of the derivative (if it exists) at
each indicated extremum
Answer:
The derivative does not exist at the extremum (-2, 0).
Step-by-step explanation:
Given function:
[tex]f(x)=(x+2)^{\frac{2}{3}}[/tex]
To differentiate the given function, use the chain rule and the power rule of differentiation.
[tex]\boxed{\begin{minipage}{5.4 cm}\underline{Chain Rule of Differentiation}\\\\If $y=f(u)$ and $u=g(x)$ then:\\\\$\dfrac{\text{d}y}{\text{d}x}=\dfrac{\text{d}y}{\text{d}u}\times\dfrac{\text{d}u}{\text{d}x}$\\\end{minipage}}[/tex]
[tex]\boxed{\begin{minipage}{5.4 cm}\underline{Power Rule of Differentiation}\\\\If $y=x^n$, then $\dfrac{\text{d}y}{\text{d}x}=nx^{n-1}$\\\end{minipage}}[/tex]
[tex]\begin{aligned}\textsf{Let}\;u &= x+2& \implies f(u) &= u^{\frac{2}{3}}\\\\\implies \dfrac{\text{d}u}{\text{d}{x}}&=1 &\implies \dfrac{\text{d}y}{\text{d}u}&=\dfrac{2}{3}u^{(\frac{2}{3}-1)}=\dfrac{2}{3}u^{-\frac{1}{3}}\end{aligned}[/tex]
Apply the chain rule:
[tex]\implies f'(x) = \dfrac{\text{d}y}{\text{d}{u}} \cdot \dfrac{\text{d}u}{\text{d}{x}}[/tex]
[tex]\implies f'(x) = \dfrac{2}{3}u^{-\frac{1}{3}} \cdot1[/tex]
[tex]\implies f'(x) = \dfrac{2}{3}u^{-\frac{1}{3}}[/tex]
Substitute back in u = x + 2:
[tex]\implies f'(x) = \dfrac{2}{3}(x+2)^{-\frac{1}{3}}[/tex]
[tex]\implies f'(x) = \dfrac{2}{3(x+2)^{\frac{1}{3}}}[/tex]
An extremum is a point where a function has a maximum or minimum value. From inspection of the given graph, the minimum point of the function is (-2, 0).
To determine the value of the derivative at (-2, 0), substitute x = -2 into the differentiated function.
[tex]\begin{aligned}\implies f'(-2) &= \dfrac{2}{3(-2+2)^{\frac{1}{3}}}\\\\ &= \dfrac{2}{3(0)^{\frac{1}{3}}}\\\\&=\dfrac{2}{0} \;\;\;\leftarrow \textsf{unde\:\!fined}\end{aligned}[/tex]
As the denominator of the differentiated function at x = -2 is zero, the value of the derivative at (-2, 0) is undefined. Therefore, the derivative does not exist at the extremum (-2, 0).
The equation 4x-10=2x represents two cars traveling in opposite directions, where x represents the distance in miles between the cars. What is the distance, in miles,between the cars?
In respοnse tο the presented questiοn, we may state that Therefοre, the equatiοn distance between the cars is 5 miles.
What is an equatiοn?Mathematicians use the term "equation" tο denοte a claim that twο expressiοns are equal. A mathematical equatiοn (=) separates each οf an equatiοn's twο sides. As an illustratiοn, the claim that the phrase "2x plus 3 equals 9" is made by the argument "2x + 3 = 9." The gοal οf equatiοn sοlving is tο identify the value οr values οf the variable(s) necessary fοr the equatiοn tο hοld true.
Equatiοns can be simple οr cοmplicated, linear οr nοnlinear, and cοntain οne οr mοre elements. The secοnd pοwer οf the equatiοn "x² + 2x - 3 = 0" is raised tο include the variable x. Mathematical disciplines like algebra, calculus, and geοmetry all make use οf lines.
The equatiοn 4x-10=2x represents twο cars traveling in οppοsite directiοns,
the distance,
[tex]$\begin{array}{c}{{4x-10=2x}}\\{{2x=10=0}}\\ {2x=10\\{x=5}}\end{array}$[/tex]
Therefore, the distance between the cars is 5 miles.
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Let X1, X2, ..., Xn denote n independent and identically distributed Bernoulli random vari- ables s.t. P(X; = 1) = p and P(Xi = 0) = 1 – p. for each i = 1, 2, ..., n. Show that __, Xi is sufficient for p by using the factorization criterion given in Theorem 9.4. THEOREM 9.4 Let U be a statistic based on the random sample Yı, Y2, ..., Yn. Then U is a sufficient statistic for the estimation of a parameter 0 if and only if the likelihood L(0) = L(y1, y2, ..., yn 10) can be factored into two nonnegative functions, L(y1, y2, ..., yn (0) = g(u,0) x h(yı, y2, ..., yn) where g(u,0) is a function only of u and 0 and h(y1, y2, ..., yn) is not a function of o.
The likelihood function can be factored using Theorem 9.4 as L(p) = L(X₁, X₂, ..., Xn | p) = g(Σⁿᵢ=1Xᵢ, p) * h(X₁, X₂, ..., Xn), where g(Σⁿᵢ=1Xᵢ, p) = p^Σⁿᵢ=1Xᵢ (1-p)^(n-Σⁿᵢ=1Xᵢ) and h(X₁, X₂, ..., Xn) = 1. This satisfies the factorization criterion, and thus, Σⁿᵢ=1Xᵢ is a sufficient statistic for p.
To show that Σⁿᵢ=1Xᵢ is sufficient for p, we need to show that the likelihood function can be factored using Theorem 9.4 as:
L(p) = L(X₁, X₂, ..., Xn | p) = g(Σⁿᵢ=1Xᵢ, p) * h(X₁, X₂, ..., Xn)
where g(Σⁿᵢ=1Xᵢ, p) is a function only of Σⁿᵢ=1Xᵢ and p, and h(X₁, X₂, ..., Xn) is not a function of p.
First, we can write the joint probability mass function of X₁, X₂, ..., Xn as:
P(X₁ = x₁, X₂ = x₂, ..., Xn = x_n) = p^Σⁿᵢ=1xᵢ (1-p)^Σⁿᵢ=1(1-xᵢ)
Taking the product of these probabilities for all i, we get:
L(p) = L(X₁, X₂, ..., Xn | p) = Πⁿᵢ=1P(Xᵢ = xᵢ) = p^Σⁿᵢ=1Xᵢ (1-p)^Σⁿᵢ=1(1-Xᵢ)
Using the factorization criterion given in Theorem 9.4, we need to find functions g(u, p) and h(X₁, X₂, ..., Xn) such that:
L(p) = L(X₁, X₂, ..., Xn | p) = g(Σⁿᵢ=1Xᵢ, p) * h(X₁, X₂, ..., Xn)
Let's take g(u, p) = pᵘ(1-p)⁽ⁿ⁻ᵘ⁾, which only depends on u and p. Then:
L(p) = L(X₁, X₂, ..., Xn | p) = g(Σⁿᵢ=1Xᵢ, p) * h(X₁, X₂, ..., Xn)
= p^Σⁿᵢ=1Xᵢ (1-p)^Σⁿᵢ=1(1-Xᵢ) * h(X₁, X₂, ..., Xn)
We can see that the term Σⁿᵢ=1Xᵢ appears in the exponent of p, and Σⁿᵢ=1(1-Xᵢ) appears in the exponent of (1-p). Therefore, we can write:
L(p) = L(X₁, X₂, ..., Xn | p) = [p^Σⁿᵢ=1Xᵢ (1-p)^Σⁿᵢ=1(1-Xᵢ)] * [1]
where the second factor is a constant function of p. This satisfies the factorization criterion, with g(u, p) = pᵘ(1-p⁽ⁿ⁻ᵘ⁾ and h(X₁, X₂, ..., Xn) = 1.
Therefore, we have shown that Σⁿᵢ=1Xᵢ is a sufficient statistic for p.
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Complete question is in the image attached below
The expression 1.05x calculates what change to the value of x?
With comparison to the initial value of x, this signifies an increase of 5%.
What is an illustration of an initial value?If, for instance, we have the mathematical problem y′=2x y ′ = 2 x, then y(3)=7 y (3) = 7 is an initial value, so when these equations are combined, they create an initial-value problem.
The calculation of a 5% increase in the value of x is done by the phrase 1.05x.
We may modify the phrase as follows to understand why:
1.05x = x + 0.05x
Thus, 1.05x is the same as increasing x by 5% from its initial value.
For example, if x = 100, then:
1.05x = 1.05 × 100 = 105
The initial value of x has increased by 5% as a result of this.
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With comparison to the initial value of x, this signifies an increase of 5%. Thus, option A is correct.
What is an illustration of an initial value?If, for instance, we have the mathematical problem y′=2x y ′ = 2 x, then y(3)=7 y (3) = 7 is an initial value, so when these equations are combined, they create an initial-value problem.
The calculation of a 5% increase in the value of x is done by the phrase 1.05x.
We may modify the phrase as follows to understand why:
1.05x = x + 0.05x
Thus, 1.05x is the same as increasing x by 5% from its initial value.
For example, if x = 100, then:
1.05x = 1.05 × 100 = 105
The initial value of x has increased by 5% as a result of this.
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Complete question:
The expression 1.05x calculates what change to the value of x?
a. increase of 5%
b. Power of 5
c. decrease of 5%
d. fraction of 5
which of these integers are pseudoprime to the base 2? multiple select question. 341 217 645 561 435
The integers that are pseudoprime to the base 2 are: 341, 217, 645, and 561. so, the correct option are A), B), C) and D).
To determine if an integer n is a pseudoprime to the base 2, we need to check if 2^(n-1) ≡ 1 mod n. If this congruence holds, then n is a pseudoprime to the base 2.
Checking each integer
341: 2^(341-1) ≡ 2^340 ≡ 1 mod 341. Therefore, 341 is a pseudoprime to the base 2.
217: 2^(217-1) ≡ 2^216 ≡ 1 mod 217. Therefore, 217 is a pseudoprime to the base 2.
645: 2^(645-1) ≡ 2^644 ≡ 1 mod 645. Therefore, 645 is a pseudoprime to the base 2.
561: 2^(561-1) ≡ 2^560 ≡ 1 mod 561. However, 561 is not a prime number, and it can be factored as 31117. Therefore, 561 is a Carmichael number, which is a composite number that satisfies the congruence condition for all bases coprime to the number. Therefore, 561 is a pseudoprime to the base 2.
435: 2^(435-1) ≡ 2^434 ≢ 1 mod 435. Therefore, 435 is not a pseudoprime to the base 2.
So, the integers are are: 341, 217, 645, and 561 that are pseudoprime to the base 2. The correct option of answer are A), B), C) and D).
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Bokomo produced 300 Granola megapacks for the Mogoditshane market. Bokomo’s marginal cost equation is as follows: MC=2x-100. Find the cost of producing an additional 200 items due to increased demand.
The cost of producing an additional 200 items due to increased demand is 180,000 Botswana Pula.
What is marginal cost?Marginal cost is the additional cost incurred by producing one additional unit of a good or service. In other words, it is the cost of producing one more unit of output.
According to question:The marginal cost (MC) equation given is MC = 2x - 100, where x is the number of units produced.
To find the cost of producing an additional 200 items due to increased demand, we need to calculate the marginal cost of producing these 200 items and then multiply that by 200.
The marginal cost of producing 200 additional items is given by:
MC(200) = 2(300 + 200) - 100
MC(200) = 2(500) - 100
MC(200) = 900
So the marginal cost of producing 200 additional items is 900. To find the total cost of producing these 200 items, we can simply multiply the marginal cost by the number of units produced, which in this case is 200:
Total cost = MC(200) × 200
Total cost = 900 × 200
Total cost = 180,000
Therefore, the cost of producing an additional 200 items due to increased demand is 180,000 Botswana Pula.
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For a certain automobile, M(x)=-.015x^2+1.32x-7.4,30=
, represents the miles per gallon obtained at a speed of x miles per hour.
(a) Find the absolute maximum miles per gallon and the speed at which it occurs.
(b) Find the absolute minimum miles per gallon and the speed at which it occurs.
Answer: (a) To find the absolute maximum miles per gallon, we need to find the maximum value of M(x). We can do this by finding the vertex of the parabola represented by M(x).
The x-coordinate of the vertex is given by:
x = -b / (2a)
where a = -0.015 and b = 1.32 in our case. Plugging in these values, we get:
x = -1.32 / (2(-0.015)) = 44
So the maximum miles per gallon occurs at a speed of 44 miles per hour.
To find the value of M(x) at this speed, we plug in x = 44:
M(44) = -0.015(44)^2 + 1.32(44) - 7.4 ≈ 32.36
Therefore, the absolute maximum miles per gallon is approximately 32.36, and it occurs at a speed of 44 miles per hour.
(b) To find the absolute minimum miles per gallon, we need to find the minimum value of M(x). We can do this by noting that the coefficient of the x^2 term is negative, which means that the parabola opens downward and has a maximum, so there is no absolute minimum.
We can also confirm this by finding the x-coordinate of the vertex, which we already calculated in part (a) to be x = 44. This means that the parabola has a minimum value of M(44), which we found to be approximately 32.36. However, this is not an absolute minimum, as there are values of M(x) that are smaller than 32.36 for other values of x. Therefore, there is no absolute minimum miles per gallon.
Step-by-step explanation:
The table represents a quadratic function.
x y
−5 −23
−4 −13
−3 −7
−2 −5
−1 −7
0 −13
1 −23
What is the equation of the function?
y = −2(x + 2)2 − 5
y = −2(x − 2)2 + 5
y = −(x + 2)2 − 5
y = −(x − 2)2 + 5
Answer:
y= - (x+2)^2-5
Step-by-step explanation:
C.
The equation of the quadratic function is y = -2x² -8x - 1.
What is quadratic equation used for?Several real-world circumstances, including projectile velocity, ball trajectory, bridge or arch design, and the behavior of specific physical and chemical processes, are modelled using quadratic functions. They are also widely utilised in physics, engineering, economics, and other disciplines.
The standard quadratic equation is given by the formula:
y = ax² + bx + c
Substituting the values from the table we have:
For (-5, -23):
25a - 5b + c = -23 .....(1)
For (-4, -13):
16a - 4b + c = -13 .......(2)
For (-3, -7):
9a - 3b + c = -7 .....(3)
Using equation 3 we have:
c = -7 - 9a + 3b
Substituting the value in equation 1 and equation 2:
25a - 5b -7 - 9a + 3b = -23
16a - 2b = -16
8a - b = -8....(4)
16a - 4b -7 - 9a + 3b = -13
7a - b = -6 ....(5)
From equation 5 we have:
b = 7a + 6
Substituting the value of b in equation 4 we have:
8a - 7a - 6 = -8
a = -2
Substitute the value of a:
b = 7(-2) + 6
b = -8
Substitute the value of a and b:
9a - 3b + c = -7
9(-2) - 3(-8) + c =-7
-18 + 24 + c = -7
6 + c = -7
c = -1
Substituting the value in standard form:
y = ax² + bx + c
y = -2x² -8x - 1
Hence, the equation of the quadratic function is y = -2x² -8x - 1.
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The relative housing cost for a US city is defined to be the ratio nationalaveragehousingcost
averagehousingcostforthecity
, expressed as a percent.
The scatterplot above shows the relative housing cost and the population density for several large US cities in the year 2005. The line of best fit is also shown and has equation y=0.0125x+61. Which of the following best explains how the number 61 in the equation relates to the scatterplot?
In 2005, even in cities with low population densities, housing costs were likely at least 61% of the national average.
We know that the relative housing cost for a US city is defined to be the ratio average housing cost for the city /national average housing cost, expressed as a percent also the scatterplot above shows the relative housing cost and the population density for several large US cities in the year 2005, therefore,
the equation is y = 0.0125x + 61
Therefore, with this we know that in 2005, even in cities with low population densities, housing costs were likely at least 61% of the national average.
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PLEASEE HELP! DUE TONIGHT
The perimeter of the figure below is 136.8 IN. Find the length of the missing side.
Show work:
Answer:
9.5in
Step-by-step explanation:
44.4in = 12.7 +22.2 +x(missing side)
34.9+x=44.4in
x=44.4in-34.9in
x=9.5in
Answer:
12.1 in
Step-by-step explanation:
what is your problem with that ? you don't understand what "perimeter" means ?
it means the whole way around the figure 1 time. it is the sum of all sides.
so, when we have the sum but miss one element of the dimensions numbers ?
what do you think we do ?
we calculate the difference between the sum of the elements we have, and the total sum.
this difference must be the missing side length.
so, the total is 136.8 in.
what we have is
44.4+10+10+12.7+22.2+12.7+12.7 =
= 44.4 + 20 + 3×12.7 + 22.2 = 124.7 in
the difference is
136.8 - 124.7 = 12.1 in
that is our missing side length.
you see, the picture tries to convince us that all the angles are right angles (90°). in that case the missing side length would be simply
44.4 - 12.7 - 22 2 = 9.5 in.
but no, it is good that we did not simply fall for an optical illusion. the absolute numbers tell us that some of the angles must be slightly different from 90°.
in case of doubt always rely on the absolute numbers.