Answer:
(f - g)(x) = -4x² + x - 8
Step-by-step explanation:
(f - g)(x) = f(x) - g(x)
= x - 8 - (4x²)
= -4x² + x - 8
A piece of wire 18cm long is bent to form a rectangle. If its length is x cm, obtain an expression for its area in terms of
* and hence calculate the dimensions of the rectangle with maximum area
Answer:
To form a rectangle, the piece of wire will have two sides of length x and two sides of length (18 - 2x)/2 = 9 - x. Therefore, the perimeter of the rectangle is given by:
2x + 2(9 - x) = 18 - 2x
The area of the rectangle is given by:
A = x(9 - x)
Expanding this expression, we get:
A = 9x - x^2
To find the dimensions of the rectangle with maximum area, we can differentiate the area expression with respect to x:
dA/dx = 9 - 2x
Setting this equal to zero to find the maximum:
9 - 2x = 0
x = 4.5
So, one side of the rectangle is x = 4.5 cm and the other side is (18 - 2x)/2 = 4.5 cm. Therefore, the dimensions of the rectangle with maximum area are 4.5 cm by 4.5 cm.
To calculate the maximum area, we can substitute x = 4.5 into the area expression:
A = 9(4.5) - (4.5)^2 = 20.25 cm^2
Step-by-step explanation:
To purchase $13200 worth of machinery for her business, Nicole made a down payment of 1200 and took out a business loan for the rest. After 3 years of paying monthly payments of 365.07, she finally paid off the loan.
(a) What was the total amount Nicole ended up paying for the machinery (including the down payment and monthly payments)?
(b) How much interest did Nicole pay on the loan?
The answer of the given question based on the compound interest to find total amount Nicole ended up paying for the machinery and the interest did Nicole pay on the loan is (A) the total amount Nicole ended up paying for the machinery is $14,342.52. (B) Nicole paid $2,342.52 in interest on the loan.
What is Compound interest?Compound interest is type of interest that is calculated not only on initial principal amount but also on accumulated interest from previous periods. In other words, interest earned in each period is added to principal amount, and interest for the next period is calculated on new, larger principal amount.
Compound interest can be thought of as "interest on interest" and is used in many financial transactions, like loans, investments, and savings accounts.
(a) The total amount Nicole ended up paying for the machinery is the sum of her down payment and all of her monthly loan payments over the 3-year period. We can calculate this as follows:
Total amount paid = Down payment + (Monthly payment x Number of payments)
Total amount paid = 1200 + (365.07 x 36)
Total amount paid = 1200 + 13142.52
Total amount paid = $14,342.52
Therefore, the total amount Nicole ended up paying for the machinery is $14,342.52.
(b) To calculate how much interest Nicole paid on the loan, we first need to calculate the total amount of the loan. We can do this by subtracting her down payment from the total cost of the machinery:
Total loan amount = Total cost of machinery - Down payment
Total loan amount = $13,200 - $1,200
Total loan amount = $12,000
Next, we can calculate the total amount of interest paid over the 3-year period by subtracting the total loan amount from the total amount paid:
Total interest paid = Total amount paid - Total loan amount
Total interest paid = $14,342.52 - $12,000
Total interest paid = $2,342.52
Therefore, Nicole paid $2,342.52 in interest on the loan.
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Michelle is baking chocolate chip cookies and she wants to cut the recipe in half. If the recipe calls for 3 cups of flour, how much flour should Michelle
add to her cookie dough?
0 1 cups of flour
O 13 cups of flour
0 1² cups of flour
0 1 cups of flour
Answer:
1 1/2 or 3/2
Step-by-step explanation:
Cutting the recipe in half means multiplying each measurement by 1/2.
3 x 1/2 = 3/2 or 1 1/2
CAN SOMEONE HELP WITH THIS QUESTION?✨
Using calculus, we can find the rate of change of area of circle and square that is 83.44 m/sec.
Define calculus?One of the most crucial areas of mathematics that addresses ongoing change is calculus. Calculus is primarily built on the two ideas of derivatives and integrals. The area under the curve of a function is measured by its integral rather than its derivative, which measures the rate of change of the function.
Whereas the integral adds together a function's discrete values over a range of values, the derivative explains the function at a particular point.
Let x be the side of the square and r be the radius of the circle,
If so, the area inside the square but outside the circle is given by:
V = Square area minus Circle area.
hence, area of a square = side² and area of a circle = π(radius)²,
Thus,
V = x² - πr²
Differentiating with respect to time (t)
dV/dt = 2x × dx/dt - 2πr dr/dt
Given,
x = 16
r = 3
dx/dt = 2 m/sec
dr/dt = 1 m/sec
⇒ dV/dt = 2 × 16 × 3 - 2π × 2 × 1
= 96 - 4π
= 96-12.56
≈ 83.44 m/sec
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A random variable X follows the uniform distribution with a lower limit of 670 and an upper limit of 770.
a. Calculate the mean and the standard deviation for the distribution. (Round intermediate calculation for Standard deviation to 4 decimal places and final answer to 2 decimal places.)
Mean:
Standard deviation:
b. What is the probability that X is less than 710? (Round intermediate calculations to at least 4 decimal places and final answer to 2 decimal places.)
Probability:
a) The mean is 720 and the standard deviation for the distribution is approximately 28.87
b. The probability that X is less than 710 is 33%
a. To calculate the mean and standard deviation, we use the following formulas:
Mean (μ) = (lower limit + upper limit) / 2
Standard deviation (σ) = (upper limit - lower limit) / √12
Substituting the values, we get:
Mean (μ) = (670 + 770) / 2 = 720
Standard deviation (σ) = (770 - 670) / √12 ≈ 28.87
b. To find the probability that X is less than 710, we use the following formula:
Probability (X < 710) = (710 - lower limit) / (upper limit - lower limit)
Substituting the values, we get:
Probability (X < 710) = (710 - 670) / (770 - 670) ≈ 0.33
This means that there is a 33% chance that X will be less than 710.
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find the equation of the line with slope 2 that goes through the point (6,1). answer using slope-intercept form.
The equation of the line with slope 2 that goes through the point (6,1) in slope-intercept form is y = 2x - 11. This means that the y-intercept of the line is -11, and the slope of the line is 2, which means that for every increase of 1 in x, the line will increase by 2 in y.
To find the equation of a line with a given slope and a point on the line, we can use the point-slope form of a linear equation:
y - y1 = m(x - x1)
where m is the slope and (x1, y1) is the given point on the line.
In this case, the slope is given as 2 and the point (6,1) is on the line. Plugging these values into the equation, we get:
y - 1 = 2(x - 6)
Expanding the right side, we get:
y - 1 = 2x - 12
Adding 1 to both sides, we get:
y = 2x - 11
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He has 2 pens. His friend gives him 2 more pens. How many pens he has?
Step-by-step explanation:
4 i guess... sry i m not good at maths
Alexander and Rhiannon left school at the same time. Alexander travelled 14 km home at an average speed of 20 km/h. Rhiannon travelled 10 km home at an average speed of 24 km/h. a) Who arrived home earlier? b) How much earlier did this person arrive at home? Give your answer to the nearest minute.
Rhiannon arrived home approximately 17 minutes earlier than Alexander.
What is the average?This is the arithmetic mean and is calculated by adding a group of numbers and then dividing by the count of those numbers. For example, the average of 2, 3, 3, 5, 7, and 10 is 30 divided by 6, which is 5.
According to the given information:To solve this problem, we can use the formula:
time = distance / speed
a) The time it took Alexander to get home is:
time_Alexander = 14 km / 20 km/h = 0.7 hours
The time it took Rhiannon to get home is:
time_Rhiannon = 10 km / 24 km/h = 0.41667 hours
Since Rhiannon's time is smaller than Alexander's, Rhiannon arrived home earlier.
b) The time difference between their arrivals is:
time_difference = time_Alexander - time_Rhiannon = 0.7 hours - 0.41667 hours = 0.28333 hours
To convert this to minutes, we can multiply by 60:
time_difference_in_minutes = 0.28333 hours x 60 minutes/hour ≈ 17 minutes
Therefore, Rhiannon arrived home approximately 17 minutes earlier than Alexander.
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b. A kennel has 90 dogs in total, some are puppies and some are adult dogs. The ratio of puppies to adult dogs in a kennel is 1:4. What fraction of dogs are adults? Use this to work out how many adult dogs there are. See the examples on the formula sheet for this assessment.
Answer:4/5 are adults. There are 72 adult dogs.
Step-by-step explanation:
a falling stone takes 0.31 s to travel past a window 2.2 m tall (fig.). from what height above the top of the window did the stone fall?
The stone fell from a height of 1.75 m above the top of the window.
We can use the kinematic equation for the vertical motion of an object in free fall to solve this problem:
y = vi*t + (1/2)at^2
where y is the initial height above the top of the window, vi is the initial velocity (which is zero since the stone is dropped), t is the time it takes for the stone to fall past the window (0.31 s), a is the acceleration due to gravity (-9.8 m/s^2).
Since the stone falls past a window that is 2.2 m tall, the final height of the stone is 2.2 m. Substituting these values into the equation, we get:
2.2 m = 0 + (1/2)(-9.8 m/s^2)(0.31 s)^2 + y
Simplifying and solving for y, we get:
y = 2.2 m - (1/2)(-9.8 m/s^2)(0.31 s)^2
y = 2.2 m - 0.45 m
y = 1.75 m
Therefore, the height from stone fell is 1.75 m.
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Graph the linear equation.
42 + 6y = -12
Plot two points on the line to graph the line.
The graph of the linear function 4x + 6y = -12 is given by the image presented at the end of the answer.
How to define a linear function?The slope-intercept representation of a linear function is given by the equation presented as follows:
y = mx + b
The coefficients of the function and their meaning are described as follows:
m is the slope of the function, representing the change in the output variable y when the input variable x is increased by one.b is the y-intercept of the function, which is the initial value of the function, i.e., the numeric value of the function when the input variable x assumes a value of 0. On a graph, it is the value of y when the graph of the function crosses the y-axis.The function for this problem is given as follows:
4x + 6y = -12.
In slope-intercept form, the function is given as follows:
6y = -4x - 12.
y = -2x/3 - 2.
The slope and the intercept are given as follows:
Intercept of b = -2, meaning that when x = 0, y = -2.Slope of -2/3, meaning that when x decays by 3, y increases by two, hence the graph also passes through point (-3,0).More can be learned about linear functions at https://brainly.com/question/24808124
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please help I know its 9:35 PM I Just need help what this question2.1 × 1.6 =
21
10
×
16
10
= tenths × tenths my parents are gonna kill me help
The value of the expression 2.1 × 1.6 = 3.36.
What are decimals?Decimals are a collection of numbers falling between integers on a number line. They are only an additional mathematical representation of fractions. Decimals allow us to express quantifiable quantities like length, weight, distance, money, etc. with more accuracy. Integers, also known as whole numbers, are represented to the left of the decimal point, while decimal fractions are shown to the right of the decimal point.
Given that the expression is: 2.1 × 1.6.
2.1 × 1.6 can be written as:
2.1 × 1.6 = 21/10 × 16/10
Multiply the numerator and denominator:
21/10 × 16/10 = 336/100
Covert the fraction into decimal:
336/100 = 3.36
Hence, the value of the expression 2.1 × 1.6 = 3.36.
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Find the measures of angles 1 through 5 in the figure shown !
Answer:
55 degrees angles on a rights angle triangle. 1 and 3 they are equal cause they are vertical opp angles 55 degrees
Participant A did 120 jumping jacks in 10 minutes. Participant B did 140 jumping jacks in 14 minutes. Which participant had the greater jumping jack rate?
Answer: Participant A
Step-by-step explanation:
if you divide 120 by 10 you would get 12 jumping jacks per minute and if you divide 140 by 14 you would get 10 jumping jacks per minute
Work out the probability of scoring a total of 4
Answer: 1/12
Step-by-step explanation:
Find the standard normal area for each of the following (LAB)Round answers to 4 decimals
The answer of the standard normal area for each of the following questions are given below respectively.
What is standard normal area?Standard normal area refers to the area under the standard normal distribution curve, which is a normal distribution with a mean of 0 and a standard deviation of 1.
a. P(1.24<Z<2.14) = 0.0912
b. P(2.03 <Z<3.03) = 0.0484
c. P(-2.03 <Z<2.03) = 0.9542
d. P(Z > 0.53) = 0.2977
Note: The standard normal distribution is a continuous probability distribution with mean 0 and standard deviation 1. The area under the curve represents probabilities and can be calculated using a standard normal distribution table or a calculator with a normal distribution function.
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Let D be the disk with center the origin and radius a. What is the average distance from points in D to the origin?
Consider a disk D having center as origin and radius 'a', the average distance from point say ( r, θ) to the origin D to the origin is equals to the 2a/3.
The distance of a point to the origin is given by the function d(x, y) = √x²+ y². Denote the disk by D. To compute the average we need to evaluate the integral, for which we use polar coordinates, f(r, θ) = (1/area of region R)∬ f(r,θ)dA.
We have a disk D with center the origin (0,0) and radius 'a'. We have to determine the average distance from points in D to the origin. Since r is defined as the distance from the origin, the distance of any point (r,θ) from the origin is f(r,θ) = r
Integrating f over the region D, with limits r = 0 to a, and 0≤ θ≤2π , dA = r×dr×dθ
[tex]∬f(r,θ)dA = \int_{0}^{a} \int_{0}^{2π} r (rdr)dθ [/tex]
[tex]= \int_{0}^{2\pi} \int_{0}^{a} r^{2} drdθ = \int_{0}^{2\pi}[\frac{{r}^{3}}{3}]_{0}^{a}dθ[/tex]
[tex]= \int_{0}^{2π}[ \frac{a³}{3} ] dθ = \frac{{a}^{3}}{3} \int_{0}^{2π}dθ[/tex]
[tex]= \frac{a³}{3} [θ]_{0}^{2π} = 2\pi(\frac{a³}{3})[/tex]
Area of disk D with radius 'a' = πa²
So, the average distance from point ( r,θ) in D to the origin is = (1/area of region D)∬f(r,θ)dA
= (1/πa²)( 2πa³/3)
= 2a/3
Hence, required distance is 2a/3.
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what is the 99% confidence interval for the difference between two means with dependent samples when the mean of the differences is d
The confidence interval for the difference between two means with dependent samples when the mean of the differences is d is given by
Confidence Interval = d ± t(0.005, n-1) × (SD / √n)
To calculate the 99% confidence interval for the difference between two means with dependent samples when the mean of the differences is d, you will need to know the standard deviation of the differences, the sample size, and the t-value for the 99% confidence level with (n-1) degrees of freedom.
Assuming that you have all these values, the formula for calculating the confidence interval is
Confidence Interval = d ± t(0.005, n-1) × (SD / √n)
Where
d is the mean of the differences between the two samples.
t(0.005, n-1) is the t-value for the 99% confidence level with (n-1) degrees of freedom. The 0.005 corresponds to the two-tailed alpha level of 0.01, since we are using a 99% confidence level.
SD is the standard deviation of the differences between the two samples.
n is the sample size.
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bug s is slow and bug f is fast. both bugs start at 0 on a number line and move in the positive direction. the bugs leave 0 at the same time and move at constant speeds. four seconds later, f is at 12 and s is at 8. how many units apart are bugs f and s thirty seconds after leaving point 0.
After 30 seconds the bugs will be apart by (x -y) = 30 units
What is speed?The speed of an object, aIso known as "v" in everyday speech and kinematics, is a scaIar quantity that measures how much its position changes over time or how much it changes per unit of time.
The distance traveIIed by an object during a time intervaI is equaI to the duration of the intervaI divided by the object's speed, with the instantaneous speed being the upper Iimit of the average speed as the intervaI's duration gets cIoser to zero. VeIocity differs from speed. Distance divided by time is the formuIa for measuring speed. A metre per second (m/s) is the metric unit of speed.
a) The speed of F is 12 units in 4 seconds, or 3 units per second.
x = 3t
The speed of S is 8 units in 4 seconds, or 2 units per second.
y = 2t
b) x-y = 3t -2t = t
After 30 seconds the bugs will be apart by (x -y) = 30 units
c) x - y = 100 = t
The bugs will be 100 units apart after 100 seconds.
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The distance between bug f and bug s after 30 seconds is: distance = 90 - 60 = 30 units.
what is algebra?
Algebra is a branch of mathematics that deals with mathematical operations and symbols used to represent numbers and quantities in equations and formulas. It involves the study of variables, expressions, equations, and functions.
When bug s and bug f leave point 0, they are initially at the same location, which is at position 0 on the number line. After four seconds, bug s has traveled a distance of 4v_s units from point 0, and bug f has traveled a distance of 4v_f units from point 0. We know that bug f has traveled 12 units in this time, so:
4v_f = 12
Solving for v_f, we get:
v_f = 3 units per second
Similarly, we know that bug s has traveled a distance of 8 units in four seconds, so:
4v_s = 8
Solving for v_s, we get:
v_s = 2 units per second
Now we can find out how far apart the bugs are after 30 seconds. Bug f has traveled a distance of 30v_f units from point 0, which is:
30v_f = 30 x 3 = 90 units
Similarly, bug s has traveled a distance of 30v_s units from point 0, which is:
30v_s = 30 x 2 = 60 units
Therefore, the distance between bug f and bug s after 30 seconds is:
distance = 90 - 60 = 30 units.
So the bugs are 30 units apart after 30 seconds.
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write the equation of a circle whose center is at latex: \left(-11,\:15\right) and whose radius is latex: 9 9 .
The equation of the circle with center at (-11, 15) and radius 9 is found to be x² + y² + 22x -30y + 337 = 0.
The center of the circle is located at the point (-11, 15). The radius of the circle is given to be 9.
Now, using the standard equation of the circle that has point (a, b) as center and r as the radius, we can write,
(x-a)² + (y-b)² = r²
Now, putting all the values,
(x-(-11))² + (y-15)² = (3)²
Solving further,
x² + 121 + 22x + y² + 225 - 30y = 9
x² + y² + 22x -30y + 337 = 0
So, the equation of the circle is found to be x² + y² + 22x -30y + 337 = 0.
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Hi please help will get max points + brainliest!
The perimeter of the given figures is: Triangle = 4x - 2. Rectangle = 8x - 8, and square = 12x - 8y.
What is perimeter?The whole distance encircling a form is referred to as its perimeter. It is the length of any two-dimensional geometric shape's border or outline. Depending on the size, the perimeter of several figures can be the same. Consider a triangle built of an L-length wire, for instance. If all the sides are the same length, the same wire can be used to create a square.
The perimeter of a figure is the sum of all the segments of the figure.
The perimeter of triangle is:
P = 2x - 5 + x + x + 3 = 4x - 2
The perimeter of rectangle is:
P = 2(l + b)
P = 2(3x + 1 + x - 5)
P = 2(4x - 4)
P = 8x - 8
The perimeter of square is:
P = 4(s)
P = 4(3x - 2y)
P= 12x - 8y
Hence, the perimeter of the given figures is: Triangle = 4x - 2. Rectangle = 8x - 8, and square = 12x - 8y.
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n january 1,2020 flounder corportaion purchased 325 of the $1000 face value, 11%, 10-year bonds of Walters Inc. The bonds mature on January 1,2030 and pay interest annually beginning January 1, 2021. Flounder purchased bonds to yield 11%. How much did Flounder pay for the bonds?
Answer:
Step-by-step explanation:
To calculate how much Flounder paid for the bonds, we need to use the present value formula for a bond:
PV = C/(1+r)^1 + C/(1+r)^2 + ... + C/(1+r)^n + F/(1+r)^n
where PV is the present value, C is the annual coupon payment, r is the yield, n is the number of years, and F is the face value.
In this case, Flounder purchased 325 bonds with a face value of $1000 each, so the total face value of the bonds is:
325 * $1000 = $325,000
The coupon rate is 11%, which means that the annual coupon payment is:
0.11 * $1000 = $110
The bonds mature in 10 years, so n = 10. The yield is also 11%, so r = 0.11.
Using these values, we can calculate the present value of the bond:
PV = $110/(1+0.11)^1 + $110/(1+0.11)^2 + ... + $110/(1+0.11)^10 + $1000/(1+0.11)^10
PV = $110/(1.11)^1 + $110/(1.11)^2 + ... + $110/(1.11)^10 + $1000/(1.11)^10
PV = $110/1.11 + $110/(1.11)^2 + ... + $110/(1.11)^10 + $1000/(1.11)^10
PV = $110*(1-(1.11)^-10)/0.11 + $1000/(1.11)^10
PV = $750.98 + $314.23
PV = $1,065.21
Therefore, Flounder paid $1,065.21 for the 325 bonds of Walters Inc.
look at this square: 2 mm 2 mm if the side lengths are tripled, then which of the following statements about its area will be true?
Therefore, the only statement that is true is: "The new area is 6 times the original area."
What is area?Area is a measure of the size or extent of a two-dimensional surface or region, such as the surface of a square, rectangle, circle, triangle, or any other shape. It is expressed in square units, such as square meters, square feet, or square centimeters.
Here,
If the side lengths of a square are tripled, the new side length will be 2 mm x 3 = 6 mm.
The original area of the square is:
Area = side length x side length = 2 mm x 2 mm = 4 mm²
The new area of the square with tripled side lengths will be:
New area = new side length x new side length = 6 mm x 6 mm = 36 mm²
Therefore, the new area of the square will be 36 mm².
To determine which of the following statements about its area will be true, we need to see which statements are true for the new area of 36 mm²:
A. The new area is 6 times the original area. This statement is true because 36 mm² is 6 times larger than 4 mm².
B. The new area is 3 times the original area. This statement is false because 36 mm² is 9 times larger than 4 mm², not 3 times larger.
C. The new area is equal to the original area. This statement is false because the new area of 36 mm² is much larger than the original area of 4 mm².
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Complete question:
look at this square: 2 mm 2 mm if the side lengths are tripled, then which of the following statements about its area will be true?
The ratio of the new area to the old area will be 2:1
The ratio of the new area to the old area will be 6:1.
The ratio of the new area to the old area will be 1:2.
The ratio of the new area to the old area will be 4:1.
Question 13 (2 points)
Suppose you flip a coin and then roll a die. You record your result. What is the
probability you flip heads or roll a 3?
1/2
3/4
7/12
1
Step-by-step explanation:
a probability is always the ratio
desired cases / totally possible cases
we have 2 possible cases for the coin and 6 possible cases for the die.
so, we have 2×6 = 12 combined possible cases :
heads, 1
heads, 2
heads, 3
heads, 4
heads, 5
heads, 6
tails, 1
tails, 2
tails, 3
tails, 4
tails, 5
tails, 6
out of these 12 cases, which ones (how many) are desired ?
all first 6 plus (tails, 3) = 7 cases
so, the correct probability is
7/12
formally that is calculated :
1/2 × 6/6 + 1/2 × 1/6 = 6/12 + 1/12 = 7/12
the probability to get heads combined with the probability to roll anything on the die, plus the probability to get tails combined with the probability to roll 3.
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PLEASE HURRY!!
Curious about people's recycling behaviors, Sandra put on some gloves and sifted through some recycling and trash bins. She kept count of the plastic type of each bottle and which bottles are properly dispensed.
What is the probability that a randomly selected bottle is correctly placed AND is a Plastic #4 bottle? Please show your work.
The probability that a randomly selected bottle is correctly placed AND is a Plastic #4 bottle is 0.25 or 25%
What is Conditional probability?
Conditional probability is the probability of an event occurring given that another event has occurred or is known to have occurred. It is denoted by P(A|B), which reads as "the probability of A given B."
The formula for conditional probability is:
P(A|B) = P(A and B) / P(B)
where P(A and B) is the probability of both events A and B occurring, and P(B) is the probability of event B occurring.
The total number of Plastic #2 bottles is 8 (correctly placed) + 5 (incorrectly placed) = 13.
The total number of Plastic #4 bottles is 5 (correctly placed) + 2 (incorrectly placed) = 7.
The probability that a randomly selected bottle is correctly placed AND is a Plastic #4 bottle is given by:
(number of Plastic #4 bottles correctly placed) / (total number of bottles)
So the probability is:
5/20 = 1/4 = 0.25
Therefore, the probability that a randomly selected bottle is correctly placed AND is a Plastic #4 bottle is 0.25 or 25%.
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Solve for the short leg of the 30-60-90 triangle.
Answer:
2
Step-by-step explanation:
The basic 30-60-90 triangle ratio is:
Side opposite the 30° angle: x
Side opposite the 60° angle: x√3
Side opposite the 90° angle: 2x
Here, the side opposite the 90° angle is 4.
This means that 2x = 4 and, thus, x = 2.
Since b is the side opposite the 30° angle, b = x, so it is 2.
For the given matrix A. find a 3 × 2 nonzero matrix B such that AB that any such matrix B must have rank 1. (Hint: The columns of B belong to the nullspace of A.] 0, Prove A= 13.421
The nullspace of matrix A found by solving the equation Ax=0. It is the nullspace of A is spanned by the vector [-2, 1, 0] and [0, -1, 1]. Any 3x2 matrix B that satisfies AB=0 belongs to the nullspace of A.
To find a 3×2 nonzero matrix B such that AB = 0, we need to find the nullspace of matrix A. The nullspace of a matrix A is the set of all vectors x such that Ax = 0.
Let's first write matrix A in row-echelon form:
1 2 1
0 -1 0
From this, we can see that the pivot variables are x1 and x2, and the free variable is x3. Thus, the general solution to Ax = 0 is given by:
x1 = -2x2 - x3
x2 = x2
x3 = x3
We can now write the columns of matrix B as:
[1, 0]
[-2, 1]
[0, -1]
To show that any such matrix B must have rank 1, we need to show that the columns of B are linearly dependent. Let's assume that B has rank 2. Then, the columns of B are linearly independent, and we can write:
c1[1, -2, 0] + c2[0, 1, -1] = [0, 0, 0]
where c1 and c2 are constants. This gives us the system of equations:
c1 = 0
-2c1 + c2 = 0
c2 = 0
which has only the trivial solution c1 = c2 = 0. This means that the columns of B are linearly dependent, and hence, any such matrix B must have rank 1.
To know more about Null space:
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_____The given question is incomplete, the complete question is given below:
For the given matrix A.= [1 2 1 1 1 1] find a 3 × 2 nonzero matrix B such that AB = 0 that any such matrix B must have rank 1. (Hint: The columns of B belong to the nullspace of A.] 0, Prove A= 13.421
cyryl hikes a distance of 0.75 kilomiters in going to school every day draw a number line to show the distance
Answer:
Step-by-step explanation:
Sure! Here's a number line showing the distance of 0.75 kilometers:
0 -------------|-------------|------------- 0.75 km
The "0" on the left represents the starting point (such as home), and the "|---|" in the middle represents the distance of 0.75 kilometers to the destination (such as school).
Scenario #3:
Imagine you find a map with a scale of 1:63,360. On that map, you see your hiking destination is
seven inches from your current location.
(a) How far away is that in reality (in miles)?
(b) Explain how you arrived at this decision.
SHOW YOUR WORK! This includes the potential for partial value, if incorrect.
Answer:
7 miles
Step-by-step explanation:
Note there are 63,360 inches in a mile.
So, rewrite scale as 1 inch: 1 mile, where inch replaces the unit.
Therefore 7 inches: 7 miles.