Answer:
Step-by-step explanation:
Choose three values for
x and substitute in to find the corresponding y values.(0,−3),(1,0),(2,3)
You can choose any of these three pairs or the equation y=3x-3
2x^3+15x^2+24x+k has 3 linear factors, two of which are identical. find k, and hence factorise the cubic
The value of k in the cubic equation is 2 and factorization of the equation is 2(x-1)^2(x+1)
What is the value of k?If 2x^3 + 15x^2 + 24x + k has three linear factors, then it must be in the form:
[tex]2x^3 + 15x^2 + 24x + k = a(x-b)(x-c)^2[/tex]
where a, b, and c are constants, and (x-c)^2 means that (x-c) is a repeated factor.
Expanding the right side, we get:
[tex]2x^3 + 15x^2 + 24x + k = a(x^3 - 2cx^2 + cx^2 - 2cx + c^2x - c^2)[/tex]
Simplifying, we get:
[tex]2x^3 + 15x^2 + 24x + k = ax^3 - 2acx^2 + ac^2x - 2acx + ac^2 + a(-c^2)\\2x^3 + 15x^2 + 24x + k = ax^3 + (-2ac + ac^2)x^2 + (-2ac + ac^2)x - ac^2[/tex]
Equating the coefficients of each power of x on both sides, we get:
[tex]a = 2\\-2ac + ac^2 = 15\\-2ac + ac^2 = 24\\-ac^2 = k\\\\[/tex]
From the second and third equations, we can see that -2ac + ac^2 = 15 and -2ac + ac^2 = 24. These equations are equivalent, so we can set them equal to each other:
[tex]-2ac + ac^2 = 15 = 24[/tex]
Simplifying, we get:
ac^2 - 2ac + 9 = 0
This is a quadratic equation in ac. Solving for ac using the quadratic formula, we get:
[tex]ac = [2 \± \sqrt(4 - 4(1)(9))] / 2[/tex]
[tex]ac = 1 \± \sqrt(-5)[/tex]
Since ac is the product of two real numbers (b and c), we know that the square root of -5 must cancel out. This is only possible if ac = 1 and c = -1. Therefore, b = -c = 1, and we can find k by setting ac^2 = -k:
[tex]1 = 2a\\15 = -2ac + ac^2 = -2a + a^2\\24 = -2ac + ac^2 = -2a + a^2\\-k = -ac^2 = 2(-1)^2 = 2\\[/tex]
Therefore, k = -2, and the factorization of 2x^3 + 15x^2 + 24x - 2 is:
2x^3 + 15x^2 + 24x - 2 = 2(x-1)^2(x+1)
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Emma and Cooper went to Tico’s tacos for lunch. Emma ordered three tacos and one burrito and Cooper ordered one taco and two burritos Emmas order total was $3.65 and Cooper’s bill was $3.30. Write and solve a system of equations to model the situation above. Explain the solution in the context of this problem. Explain, or show your work in the box below.
In the given system of equations one taco costs $0.80 and one burrito costs $0.72.
What is a system of equations?An equation system is a finite collection of equations for which we searched for the common solutions. It is sometimes referred to as a set of simultaneous equations or an equation set. The classification of a system of equations is similar to that of a single equation. In modelling issues where the unknown values may be expressed in the form of variables, a system of equations finds use in everyday life.
Let us suppose the cost of one taco = x.
Let us suppose the cost of one burrito = y.
Then, for Emma we have:
3x + y = 3.65
For Cooper we have:
x + 2y = 3.30
Using elimination, multiply the first equation by 2 and subtract it from the second equation:
(2)(3x + y = 3.65)
6x + 2y = 7.30
x + 2y = 3.30
-5x = -4
x = 4/5
Substituting this value of x into either equation:
3(4/5) + y = 3.65
y = 2.15/3 ≈ 0.72
Therefore, one taco costs $0.80 and one burrito costs $0.72.
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Grace wants to buy a jump rope that costs $7, a board game that costs $10, and a playground ball that costs $4. She has saved $10 from her allowance, and her uncle gave her $3. How much more money does Grace need to buy the jump rope, the game, and the ball?
Grace need to buy the jump rope, the game, and the ball $8.
$7 will get you a jump rope.
$10 will get you a board game.
$4 will get you a playground ball.
total amount to be spent: $7 + $10 + $4 = $21
She has ten dollars.
$3 was all her uncle gave her.
13 dollars are all she has.
She needed $8, therefore 21 - 13 = $8.
a sum of money awarded as compensation, a bounty, or to cover costs. a wage that comes with a cost-of-living supplement. especially: a regular amount set aside for household or personal costs.
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use a direct proof to show that every odd integer is the difference of two squares. [hint: find the difference of the squares of k 1 and k where k is a positive integer.]
Yes, every odd integer can be written as the difference of two squares.
To prove this, let k be a positive integer. Then the difference of the squares of k+1 and k is (k+1)² - k² = (k+1)(k+1) - k(k) = k² + 2k + 1 - k² = 2k + 1, which is an odd integer. Thus, every odd integer can be written as the difference of two squares.
To prove this, we first chose a positive integer, k. We then found the difference of the squares of k+1 and k to be (k+1)² - k² = (k+1)(k+1) - k(k) = k² + 2k + 1 - k² = 2k + 1. Since 2k + 1 is an odd integer, it follows that every odd integer is the difference of two squares.
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The monthly rent for a pizza parlor is $1,200. the average production cost per pizza is $6.75. the monthly expenses for the pizza parlor are given by the function E(x) = 1,200 + 6.75x, where x is the number of pizzas sold. for x pizzas sold, the pizza parlor's revenue is given by the function R(x) = 12.5x. the monthly profit of the pizza parlor is the difference between its revenue and its expenses. which function represents the monthly profit, P(x)?
The monthly profit, P(x), of a pizza parlor is given by the function P(x) = 5.75x - 1,200, where x is the number of pizzas sold, given revenue and expense functions.
The monthly profit, P(x), is the difference between the revenue and the expenses, so:
P(x) = R(x) - E(x)
From the given information:
R(x) = 12.5x (revenue per pizza is $12.5, and x is the number of pizzas sold)
E(x) = 1,200 + 6.75x (expenses, which include the monthly rent of $1,200 and the production cost of $6.75 per pizza, multiplied by the number of pizzas sold)
Substituting the values of R(x) and E(x) into the equation for P(x):
P(x) = 12.5x - (1,200 + 6.75x)
Simplifying the equation:
P(x) = 5.75x - 1,200
Therefore, the function representing the monthly profit, P(x), is P(x) = 5.75x - 1,200.
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Answer:
P(x)= 5.75x -1,200
Step-by-step explanation:
Plato/Edmentum
Drag the correct equation for each graph into the blank spaces. Remember you should convert to slope intercept
form first!
In both cases, the slope (m) is the coefficient of the x-variable, and the y-intercept (b) is the constant. In the first equation, the slope (m) is 2, and the y-intercept (b) is -3. In the second equation, the slope (m) is 3, and the y-intercept (b) is 6.
What is graph?Graph is a data structure that consists of nodes (vertices) and edges (connections between nodes). Graphs are used to represent networks, such as social networks, transportation networks, and communication networks. Graphs provide a way to visualize relationships between data points and can be used to discover patterns in data and make predictions. Graphs are a powerful tool in data science and can be used to analyze complex systems.
This is happening because the equations represent the linear relationships between the variables of the graphs. The first equation is in slope-intercept form, and can be used to calculate the y-value of a given x-value. The equation for the second graph is also in slope-intercept form, and can be used to calculate the y-value of a given x-value.
In both cases, the slope (m) is the coefficient of the x-variable, and the y-intercept (b) is the constant. In the first equation, the slope (m) is 2, and the y-intercept (b) is -3. In the second equation, the slope (m) is 3, and the y-intercept (b) is 6.
These equations represent the linear relationships between the variables of the graphs. The slope of the line is the rate of change between the x- and y-values, while the y-intercept is the starting point of the line. By using the equations, we can calculate the y-value of any given x-value on the graph, and thus accurately represent the linear relationships between the variables.
The diagram is given below.
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Complete questions as follows-
When solving the equation 3(x-2)=-12, what is a possible first step?
*
1)Distributing the 3 into each term of the parenthesis.
2)She could either divide by 3 or distribute 3 into the parenthesis
3)Adding 2 to each side of the equation
4) Subtracting 2 on each side of the equation
5) Dividing each side of the equation by 3
6)none of the answer choices tell what the first step could be.
Answer:
2)She could either divide by 3 or distribute 3 into the parenthesis
Which of the following Boolean expressions are equivalent to the expression num ≥ 15 ?Select two answers.(A) (num > 15) AND (num = 15)(B) (num > 15) OR (num = 15)(C) NOT (num < 15)(D) NOT (num < 16)
Boolean expressions that are equivalent to the expression num ≥ 15 are (B) (num > 15) OR (num = 15) and (D) NOT (num < 16).
In Boolean algebra, a Boolean expression is an algebraic expression made up of Boolean variables and logical operations. Boolean logic deals with logical variables and has only two values: 1 or 0 or TRUE or FALSE. Boolean expressions evaluate to a Boolean value of either 0 or 1 or TRUE or FALSE.(A) (num > 15) AND (num = 15) - this expression is not equivalent to num ≥ 15 because (num > 15) is false for all values of num that are less than or equal to 15.
However, num ≥ 15 is true for all values of num that are greater than or equal to 15.(C) NOT (num < 15) - this expression is not equivalent to num ≥ 15 because NOT (num < 15) is true for all values of num that are greater than or equal to 15. However, num ≥ 15 is true for all values of num that are greater than or equal to 15.(B) (num > 15) OR (num = 15) - this expression is equivalent to num ≥ 15 because if num is greater than 15, then (num > 15) is true and (num = 15) is false. However, if num is equal to 15, then (num > 15) is false and (num = 15) is true.
In either case, (num > 15) OR (num = 15) is true, which means that num ≥ 15 is true.(D) NOT (num < 16) - this expression is equivalent to num ≥ 15 because if num is less than 16, then (num < 16) is true and NOT (num < 16) is false. However, if num is greater than or equal to 16, then (num < 16) is false and NOT (num < 16) is true. In either case, NOT (num < 16) is true, which means that num ≥ 15 is true.
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80% off of the sale price $90 what is the original price
Answer: So that means the answer is $112.5
Step-by-step explanation:Percent of Discount is 80%. Sale Price is $90. The original price,. = 90 x 100 / 80. = 9000/80. = 112.5. Therefore, $112.5 is the original price.
YES THIS IS RIGHT!!!!
Gastonia’s population of 15,000 in 2020 was expected to grow exponentially by 12% each DECADE for the rest of the century…
-------------------------------------------------
What will be the population in 2030?
ANSWER: people
-------------------------------------------------
What will be the population in 2040?
ANSWER: people
Answer: In 2030, the population will be 16,800 and in 2040 the population will be 18,816.
Step-by-step explanation: To find the population in 2030, we need to calculate one decade of growth from 2020 to 2030:
Population in 2030 = 15,000 x (1 + 0.12)^1 = 16,800
Therefore, the population in Gastonia is expected to be 16,800 in 2030.
To find the population in 2040, we need to calculate two decades of growth from 2020 to 2040:
Population in 2040 = 15,000 x (1 + 0.12)^2 = 18,816
Therefore, the population in Gastonia is expected to be 18,816 in 2040.
write subtraction problem as an addition problem problem she writes what is -3+ -4 how can how can you use a number right now -3+ -4 as a subtraction problem
the subtraction problem "-3 - 4" is equivalent to the addition problem "-3 + (-4)".
How to solve and what is subtraction?
To write the subtraction problem "-3 - 4" as an addition problem using negative numbers, we can rewrite it as follows:
-3 - 4 = -3 + (-4)
So, the subtraction problem "-3 - 4" is equivalent to the addition problem "-3 + (-4)".
Subtraction is a fundamental arithmetic operation used to find the difference between two values or quantities. It involves taking away a certain amount from a starting value, resulting in a lower value.
The starting value is called the minuend, the amount being subtracted is called the subtrahend, and the result is called the difference. Subtraction is commonly used in everyday life, such as in calculating change when making a purchase or determining how much time has elapsed between two events. It is also an important concept in more advanced mathematical topics such as algebra and calculus.
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Write an equation for an ellipse centered at the origin, which has foci at (\pm\sqrt{12},0)(± 12 ,0)left parenthesis, plus minus, square root of, 12, end square root, comma, 0, right parenthesis and vertices at (\pm\sqrt{37},0)(± 37 ,0)left parenthesis, plus minus, square root of, 37, end square root, comma, 0, right parenthesis
The equation for the ellipse is: [tex]$\frac{x^2}{37}[/tex] + [tex]$\frac{y^2}{25}[/tex] [tex]= 1$$[/tex]
The standard equation for an ellipse centered at the origin is:
[tex]$\frac{x^2}{a^2}[/tex] + [tex]$\frac{y^2}{b^2}[/tex] [tex]= 1$$[/tex]
where a is the distance from the center to a vertex, and b is the distance from the center to a co-vertex.
In this case, the vertices are located at [tex]$(\pm\sqrt{37}, 0)$[/tex], which means [tex]$a=\sqrt{37}$[/tex]. The distance between the foci is [tex]$2c=2\sqrt{12}=2\sqrt{3\times 4}=2\sqrt{3}\times 2=4\sqrt{3}$[/tex], which means [tex]$c=2\sqrt{3}$[/tex].
The value of b can be found using the relationship between a, b, and c in an ellipse:
[tex]$$a^2 = b^2 + c^2$$[/tex]
Substituting the values we know, we get:
[tex]$$37 = b^2 + (2\sqrt{3})^2$$[/tex]
Simplifying:
[tex]$$37 = b^2 + 12$$[/tex]
[tex]$$b^2 = 37 - 12 = 25$$[/tex]
Taking the square root of both sides, we get:
[tex]$$b = \pm 5$$[/tex]
Since the co-vertices are located at [tex]$(0,\pm b)$[/tex], we can see that [tex]$b=5$[/tex] (and not -5, since the ellipse is centered at the origin).
Therefore, the equation for the ellipse is:
[tex]$\frac{x^2}{37}[/tex] + [tex]$\frac{y^2}{25}[/tex] [tex]$ = 1$$[/tex]
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Question In an accounting college class, 50% of students receive a B or above on the final exam. 84 students are randomly selected from an accounting class at a local college. Let X be the number of students who received a B or above on the final exam. What normal distribution best approximates X? • Round to one decimal place if entering a decimal answer below.
The normal distribution that best approximates X is N(42, 21).
In an accounting college class, 50% of students receive a B or above on the final exam. 84 students are randomly selected from an accounting class at a local college. Let X be the number of students who received a B or above on the final exam. The normal distribution that best approximates X is a normal distribution with mean µ = np and variance σ^2 = np(1 - p).
In this case, n = 84 and p = 0.5 since 50% of the students receive a B or above on the final exam, and we want to find the distribution of the number of students who receive a B or above on the final exam.X ~ N(µ, σ^2) = N(np, np(1 - p)) = N(84(0.5), 84(0.5)(0.5)) = N(42, 21)
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what is the area of the parallelogram
The area of the parallelogram is 324 square yards.
What is a parallelogram ?
A parallelogram is a four-sided geometric shape that has two pairs of parallel sides. It isdefined by its four vertices, four sides, and two diagonals that intersect at their midpoint. The opposite sides of a parallelogram are congruent, and the opposite angles are also congruent. The adjacent angles are supplementary and add up to 180 degrees.
To find the area of a parallelogram, we need to multiply the length of the base by the height, which is the perpendicular distance from the base to the opposite side. This formula is similar to finding the area of a rectangle, where the base is one side of the rectangle and the height is the distance from that side to the opposite side.
Calculating the area of the given parallelogram :
The area of a parallelogram is A = bh, where b is the base and h is the height.
Given the height of the parallelogram is 12 yards and the base is 27 yards. Using the formula for the area of a parallelogram, we can calculate the area as follows:
A = bh
A = 27 yards × 12 yards
A = 324 square yards
Therefore, the area of the parallelogram is 324 square yards.
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What is the perimeter of the trapezoid below?
Answer:
82
Step-by-step explanation:
divide this shape into a triangle and rectangle and use 24 as the base of the triangle using the Pythagoras theorem you get that the perpendicular is 7 and since 13 is also parallel to that side, 13+7=20 so one side equals 20 one side equals 25 one side equals 24 and one side equals 13 now add all of these to get 82
Step-by-step explanation:
See image below ..... perimeter = 25 + 13 + 24 + 13 + x
x is found using pythagorean theorem for right triangles
x = 7 units
total perimeter is then 82 units
What is 999999999999999999292929292929923798341038701384710348710387017+74569273469287465294875629837469523746539287465927346592873654987264982768
Answer: 8377828766289
Step-by-step explanation:
Answer:
The sum is:
74569273469287465294875629837469523746539287465927346592873654987264982768
999999999999999999292929292929923798341038701384710348710387017
= 74569273469287465294875629837469523746539287465927346592873655087264969785
What is −10y − 19 if y is 9?
Answer: -109
Step-by-step explanation:
Concept : First take note of what operations are in play and how they relate to y. In this case, -10 is being multiplied to y and then 19 is subtracted from the product of -10 and y.
Steps:
Plug 9 in for y (parentheses just mean multiply):
-10(9)-19
Simplify using correct order of operations:
-90-19
-109
Hope this helps :)
indicate the method you would use to prove the two triangles ~=. if no method applies, enter none
The method we would be using here to prove the triangle's congruency is LA, the leg-acute theorem.
Define congruency?The dimensions of the sides and angles of two or more triangles determine whether they are congruent or not. A triangle's size and shape are determined by its three sides and three angles, respectively. If pairings of corresponding sides and corresponding angles are equal, two triangles are said to be congruent. They are the exact same size and form.
Here in the given figures, we can use the LA theorem also known as leg acute. It says that a right triangle is congruent if its leg and any acute angle are congruent with the corresponding leg and any acute angle of another right triangle.
The hypotenuse of the first triangle is corresponding to the hypotenuse of the second triangle.
The angle opposite to the right angle in both the triangles are corresponding to each other.
So as per the LA theorem we can prove that the triangle is congruent.
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a pastry chef accidentally inoculated a cream pie with six s. aureus cells. if s. aureus has a generation time of 60 minutes, how many cells would be in the cream pie after 7 hours?
After the time of seven hours, the cream pie would have approximately 768 S. aureus cells after 7 hours with a generation time of 60 minutes.
How many cells would be in the cream pie after 7 hours?Six S. aureus cells have been accidentally inoculated into a cream pie. S. aureus has a generation time of 60 minutes. S. aureus is a pathogenic bacterium found in the environment, as well as on the skin, and in the upper respiratory tract.
The generation time of this bacterium is 60 minutes, meaning that a single bacterium can produce two new cells in 60 minutes.
If there are 6 S. aureus cells in a cream pie, the number of bacteria will continue to increase as time passes.
The number of generations (n) in seven hours is calculated as:
n = t/g
n = 7 hours × 60 minutes/hour/60 minutes/generation = 7 generations
The number of cells in the cream pie after 7 hours is calculated as :
N = N₀ × 2ⁿ
N = 6 cells × 2⁷
N = 768 cells
Therefore, after seven hours, the cream pie would have approximately 768 S. aureus cells.
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The function g is continuous on the interval [a, b] and is differentiable on (a, b). Suppose that g(x) = 0 for 4 distinct values of x in (a, b). What is the minimum number, k, of z in (a, b) such that g'(z) = 0?
We have to find the minimum number, k, of z in (a, b) such that g'(z) = 0. The function g is continuous on the interval [a, b] and is differentiable on (a, b). Suppose that g(x) = 0 for 4 distinct values of x in (a, b).
Let x1, x2, x3, and x4 be the four distinct values of x such that g(x) = 0.Now consider the following cases:Case 1: All four x1, x2, x3, x4 are local extrema of g(x).If this is the case, then g′(x1)=g′(x2)=g′(x3)=g′(x4)=0. Therefore, the minimum number, k, of z in (a, b) such that g′(z) = 0 is 4.Case 2:
There are less than four local extrema of g(x).In this case, by Rolle's Theorem, there exists at least one point z in (a, b) such that g′(z)=0. Since there are less than four local extrema of g(x), this point z is not equal to any of x1, x2, x3, and x4. Therefore, the minimum number, k, of z in (a, b) such that g′(z) = 0 is 1.In conclusion, the minimum number, k, of z in (a, b) such that g′(z) = 0 is either 1 or 4 depending on whether there are less than four local extrema of g(x) or all four x1, x2, x3, and x4 are local extrema of g(x).
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A triangle has a base that measures 6.4 feet. Its height measures 5 feet. How many square feet is the area of the triangle?
Answer:
16 feet
Step-by-step explanation:
Area of triangle = 1/2(a x b)
A = 1/2(6.4 x 5)
A = 32/2
A = 16 feet
In an introductory psychology class with n = 50 students, there are 9 freshman males, 15 freshman females, 8 sophomore males, 12 sophomore females, and 6 junior females. A random sample of n = 2 students is selected from the class. If the first student in the sample is a male, what is the probability that the second student will also be a male?
a. 8/14
b. 12/20
c. 20/44
d. 20/50
Answer: Total number of males= C
Step-by-step explanation:
PLEASE HELP ME
(Answer these four questions please)
3. The statement is true, the correlation coefficient is close to -1. 4. temperature for a city with a latitude of 48 is 43. 5. The statement is false. 6. cannot make a reasonable estimate
Describe Equation?Equations can be used to model real-world situations and solve problems in many fields, including science, engineering, finance, and more. They are an essential tool in mathematics and are used extensively in algebra, calculus, and other advanced branches of math.
Equations can involve various mathematical operations such as addition, subtraction, multiplication, division, exponentiation, and others.
Question 3:
The statement is true. We can check this by calculating the correlation coefficient between the latitude and temperature data points, which should be close to -1. The calculated line of best fit is also consistent with the given data.
Question 4:
To estimate the temperature for a city with a latitude of 48, we can use the equation of the line of best fit:
y = -1.07x + 92.87
Substituting x = 48, we get:
y = -1.07(48) + 92.87
y = 42.79
Rounding to the nearest whole number, the estimated temperature for a city with a latitude of 48 is 43.
Question 5:
The statement is false. We can check this by calculating the correlation coefficient between the passengers and suitcases data points, which should be close to 1. The given line of best fit has a negative slope, which is inconsistent with the positive correlation between the variables.
Question 6:
To estimate the number of suitcases for a flight carrying 250 people, we can use the equation of the line of best fit:
y = -1.98x + 7.97
Substituting x = 250, we get:
y = -1.98(250) + 7.97
y = -485.03
However, it does not make sense for the number of suitcases to be negative. Therefore, we cannot make a reasonable estimate for the number of suitcases on a flight carrying 250 people using this line of best fit.
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1. Find the length of an arc of a circle with radius 21 m that subtends a central angle of 15°
The length of the arc is approximately 5.51 meters when a circle with a radius of 21 meters is subtended by a central angle of 15 degrees.
The length of an arc of a circle with radius 21m that subtends a central angle of 15° can be calculated using the formula:
Arc length = (central angle/360°) x 2πr
where r is the radius of the circle, and π is the mathematical constant pi.
Substituting the given values, we get:
Arc length = (15/360) x 2π x 21
Arc length = (1/24) x 2 x 3.14 x 21
Arc length = (1/12) x 3.14 x 21
Arc length = 5.51 meters (rounded to two decimal places).
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We need to apply the formula to determine the length of a circle's arc: (Central angle / 360°) x (2 x x radius) is the formula for arc length. the radius is distance from the circle center to any point on its perimeter,
and the central angle is the angle subtended by the arc at its center. The radius in this instance is stated as 21 meters, while the arc's center angle is provided as 15 degrees. When these values are added to the formula, we obtain: arc length is equal to (15°/360°) x (2x x 21m) 3.68 m. As a result, the arc measures around 3.68 meters in length. As a result, the radius is the distance from the circle's center to any point on its perimeter, if we were to sketch an arc of The arc's length would be around 3.68 meters for a circle with a radius of 21 meters and a center angle of 15 degrees.
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If John had 3 apples then Droped 2 then found 4 then gave one to his friend how many apples does he have now
Answer:
4 apples
Step-by-step explanation:
We know
John had 3 apples, then Dropped 2, found 4, then gave 1 to his friend.
How many apples does he have now?
3 - 2 + 4 - 1 = 4 apples
So, he has 4 apples now.
Answer:
4 apples
Step-by-step explanation:
We know that John had 3 apples
but then he had dropped 2 of them
he then found 4
then he gave 1 to his friend
3-2=1
1+4=5
5-1=4
The answer is 4
Hope this helps!
Can anyone solve this ???
The result (recurrent value), A = sum j=1 to 89 ln(j), is true for every n. This is the desired result.
How do you depict a relationship of recurrence?As in T(n) = T(n/2) + n, T(0) = T(1) = 1, a recurrence or recurrence relation specifies an infinite sequence by explaining how to calculate the nth element of the sequence given the values of smaller members.
We can start by proving the base case in order to demonstrate the first portion through recurrence. Let n = 1. Next, we have:
Being true, ln(a1) = ln(a1). If n = k, let's suppose the formula is accurate:
Sum j=1 to k ln = ln(prod j=1 to k aj) (aj)
Prod j=1 to k aj * ak+1 = ln(prod j=1 to k+1 aj)
(Using the logarithmic scale) = ln(prod j=1 to k aj) + ln(ak+1)
Using the inductive hypothesis, the property ln(ab) = ln(a) + ln(b)) = sum j=1 to k ln(aj) + ln(ak+1) = sum j=1 to k+1 ln (aj)
(b), we can use the just-proven formula:
A = ln(1, 2,...) + ln + ln (89)
= ln(j=1 to 89) prod
sum j=1 to 89 ln = ln(prod j=1 to 89 j) (j).
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A 10 kilogram object suspended from the end of a vertically hanging spring stretches the spring 9.8 centimeters. At time t = 0, the resulting mass-spring system is disturbed from its rest state by the force F(t) = 100cos(8t). The force F(t) is expressed in Newtons and is positive in the downward direction, and time is measured in seconds.
Determine the spring constant k.
k = ? Newtons / meter
Formulate the initial value problem for y(t), where y(t) is the displacement of the object from its equilibrium rest state, measured positive in the downward direction. (Give your answer in terms of y,y′,y′′,t.)
Differential equation: ?
Initial conditions: y(0) = ? and y′(0) = ?
Solve the initial value problem for y(t).
y(t) = ?
Plot the solution and determine the maximum excursion from equilibrium made by the object on the time interval 0 ≤ t < [infinity]. If there is no such maximum, enter NONE.
maximum excursion = ? meters
Using Hooke's law the maximum value of |cos(8t)| is 1, so the maximum excursion is 1/6 meters.
To find the spring constant k, we use Hooke's law:
F = -ky
where F is the weight of the object, and y is the distance it is stretched from its rest position. At equilibrium, F = mg = 10 × 9.81 = 98.1 N. Thus,
98.1 = -k × 0.098
k = -1000 N/m
The equation of motion for the system is given by:
my'' + ky = F(t)
Substituting the given values, we get:
10y'' + (-1000)y = 100cos(8t)
y'' - 100y = 10cos(8t)
with initial conditions y(0) = 0 and y'(0) = 0.
The characteristic equation is r² - 100 = 0, with roots r = ±10i. The complementary solution is therefore y_c(t) = c1cos(10t) + c2sin(10t).
For the particular solution, we assume a form of yp(t) = Acos(8t) + Bsin(8t), and substitute it in the differential equation to get:
-64Acos(8t) - 64Bsin(8t) - 100(Acos(8t) + Bsin(8t)) = 10cos(8t)
Solving for A and B, we get A = -1/6 and B = 0. Thus, the particular solution is yp(t) = (-1/6) × cos(8t).
The general solution is therefore y(t) = c1cos(10t) + c2sin(10t) - (1/6)*cos(8t). Applying the initial conditions, we get c1 = 0 and c2 = 0, so the particular solution is simply y(t) = (-1/6) × cos(8t).
The maximum excursion from equilibrium can be found by taking the absolute value of y(t) and finding its maximum value. We have:
|y(t)| = (1/6) × |cos(8t)|
The maximum value of |cos(8t)| is 1, so the maximum excursion is 1/6 meters.
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assistants please --------> The members of a weight loss support group just collectively weighed in at 1,519 pounds, which is 2% lighter than their previous weight. How much did the group weigh last time?
Work Shown:
x = previous total weight
2% of x = 0.02x = amount of weight lost
x - 0.02x = 0.98x = current total weight
0.98x = 1519
x = 1519/0.98
x = 1550 pounds
Check:
2% of 1550 = 0.02*1550 = 31
The group collectively lost 31 pounds.
1550-31 = 1519
The answer is confirmed.
Note that 98% of 1550 = 0.98*1550 = 1519
You pick a card at random. 1 2 3 What is P(not even)?
An 1 οr perhaps an even integer will be drawn back 75% οf the times frοm the set οf 1, 2, 3 and 4. Thus, Prοbability P(nοt even) is 75%.
Hοw simple is prοbability?Prοbability is the likelihοοd that sοmething will οccur οr the prοbability that sοmething will happen. Prοbability is the measure οf hοw prοbable it is that a cοin will land heads up after being tοssed intο the air.
As prοbabilistic arguments sοmetimes prοduce οutcοmes that appear cοntradictοry οr illοgical, prοbability is usually regarded as amοng the mοst challenging tοpics οf mathematics.
P(1) = 1/4(there is one card with a 1)
P(even) = 2/4 = 1/2 (there are 2 cards with even numbers out of 4)
Therefore,
P( 1 or even) =P(1) + P(even)
= 1/4 + 1/2
= 3/4
To express the answer as a percentage, we can multiply by 100:
P(1 or even) = 3/4 × 100%
=> 75%
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You pick a card at random. card 1, card 2 and card 3
What is P(not even)?
Write your answer as a fraction or percentage.
a red cap fire hydrant provides 1700 liters per minute of water. how long (in minutes to the nearest minute) will it take to fill a water truck with a tank of the following dimensions: 68 inches diameter and 24 feet long? when the tank is full of water, how heavy will the water load be in pounds (lbm) to the nearest pound?
It will take approximately 8 minutes to fill the water truck, and the weight of the water load will be approximately 30,296 pounds.
How to find out how long it will take to fill the tank.First, we need to convert the dimensions of the tank from inches to feet:
68 inches diameter = 68/12 feet = 5.67 feet diameter
24 feet long = 24 feet long
Next, we can calculate the volume of the tank in cubic feet:
Volume = [tex]pi x (diameter/2)^2 x length[/tex]Volume = [tex]3.14 x (5.67/2)^2 x 24[/tex]Volume = [tex]485.15 cubic feet[/tex]Since 1 cubic foot of water weighs 62.4 pounds, we can calculate the weight of the water in the tank in pounds:
Weight = Volume x DensityWeight = 485.15 x 62.4Weight = 30,296.16 poundsTo find out how long it will take to fill the tank, we can use the flow rate of the fire hydrant:
Flow rate = 1700 liters per minute1 liter = 0.264172 gallonsFlow rate = 1700 x 0.264172 = 449.10 gallons per minute1 gallon = 0.133681 cubic feetFlow rate = 449.10 x 0.133681 = 60.05 cubic feet per minuteFinally, we can divide the volume of the tank by the flow rate to find out how long it will take to fill the tank:
Time = Volume / Flow rateTime = 485.15 / 60.05Time = 8.08 minutesTherefore, it will take approximately 8 minutes to fill the water truck, and the weight of the water load will be approximately 30,296 pounds.
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