To find the length of a'b', we first need to know the scale factor of the dilation. The scale factor is given by the ratio of the corresponding side lengths in the original and diluted figures.
In this case, we are given that the original figure Aabc has been diluted by a factor of √2. So the length of each side in the dilated figure aa'b'c is √2 times the length of the corresponding side in Aabc.
To find the length of a'b, we can use the Pythagorean theorem in the right triangle aa'b'. Since we know that ab is one of the legs of this triangle, we can find its length as follows:
ab = (a'b' / √2) * sin(28°)
We are not given the length of ab or a in the original figure, so we cannot find their exact values. However, we can find the measure of angle A using the Law of Sines in triangle Aab:
sin(A) / ab = sin(62°) / b
where b is the length of side bc in Aabc. Solving for sin(A) and substituting the expression for ab that we found earlier, we get:
sin(A) = (sin(62°) / b) * [(a'b' / √2) * sin(28°)]
Since we know the values of sin(62°) and sin(28°), we can simplify this expression and use a value for b (if it is given in the problem) to find sin(A) and then A.
Use the nth-term test for divergence to show that the series is divergent, or state that the test is inconclusive.
∑ 1/n+13
Using the nth-term test for divergence on the series ∑ 1/n+13 is inconclusive. However, by comparing the series to the divergent harmonic series, we can conclude that ∑ 1/n+13 is also divergent.
We can use the nth-term test for divergence to determine the convergence or divergence of the series
lim n → ∞ (1/n+13) = 0
Since the limit of the nth term is 0, the nth-term test is inconclusive, and we cannot determine the convergence or divergence of the series using this test.
However, we can use the comparison test to show that the series diverges. We can compare the given series to the harmonic series, which we know diverges
1/1 + 1/2 + 1/3 + ...
Since each term of the given series is less than the corresponding term of the harmonic series, the given series must also diverge. Therefore, the series ∑ 1/n+13 is divergent.
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(2) Use the Comparison Test or the Limit Comparison Test to determine the convergence or divergence of the following series. Justify your answer. 23-1 5k - 1 k=1 (3) Use the Integral Test to determine the convergence or divergence of the following series. Justify your answer. Ink k k=1
2)By using Comparison Test,the series 23-1 5k-1 k=1,is divergent.
3)By using Integral Test the series Ink k k=1 is divergent.
2) To determine the convergence or divergence of the series 23-1 5k - 1 k=1:
For the first series, 23-1 5k-1 k=1, we can use the Limit Comparison Test.
Let's compare it to the series 5k-1 k=1.
We take the limit as k approaches infinity of the ratio of the two series:
lim(k->∞) [(23-1 5k-1) / (5k-1)] = lim(k->∞) [23 / 5] = 23/5
Since this limit is finite and positive, and the series 5k-1 diverges (as it is a p-series with p=1),
we can conclude that the given series also diverges.
3)To determine the convergence or divergence of the series Ink k k=1:
For the second series, Ink k=1, we can use the Integral Test.
We need to check if the following improper integral converges or diverges:
∫(1 to ∞) ln(x) dx
Integrating by parts, we get:
∫(1 to ∞) ln(x) dx = [xln(x) - x]1∞ + ∫(1 to ∞) dx/x
The first term evaluates to -∞, and the second term is the divergent harmonic series.
Therefore, the improper integral and the series both diverge.
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Nine hundred thirty six student's, 65% of the entire student body, attended the football game. find the size of the student body.
The size of the student body is approximately 1440 students.
To determine the size of the student body, we'll use the given information that 936 students represent 65% of the total number of students. We can set up a proportion to solve for the unknown total (let's call it "x"):
(65% of x) = 936
To express the percentage as a decimal, divide 65 by 100, which equals 0.65:
0.65 * x = 936
Next, to find the value of x, divide both sides of the equation by 0.65:
x = 936 / 0.65
x ≈ 1440
So, the size of the student body is approximately 1440 students. In this problem, we used the concept of percentage to find out the total number of students in the student body, knowing that 936 students (65%) attended the football game.
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Suppose a mouse is placed in the maze at the right. if each desicion about direction is made at random, create a simulation to determine the probability that the mouse will find its way out before coming to a dead end or going out in the opening.
The probability of the mouse finding its way out before reaching a dead end or going out in the opening can be estimated by dividing the number of successful outcomes by the total number of trials in the sample.
To create a simulation to determine the probability that the mouse will find its way out before coming to a dead end or going out in the opening, we can follow these steps:
Create a model of the maze in a programming language such as Python.Define the starting position of the mouse as the position on the right side of the maze.Define the exit position of the maze as the position on the left side of the maze.Randomly choose a direction for the mouse to move in (up, down, left or right).Check if the chosen direction leads to a dead end or out of the maze. If it does, return a failure outcome.If the chosen direction leads to a viable path, move the mouse to that position and repeat steps 4-6 until the mouse either reaches the exit or gets stuck in a dead end.Repeat steps 2-6 multiple times to generate a sufficient sample size.Calculate the proportion of successful outcomes (i.e. the mouse finding its way out before reaching a dead end or going out in the opening) from the generated sample.The probability of the mouse finding its way out before reaching a dead end or going out in the opening can be estimated by dividing the number of successful outcomes by the total number of trials in the sample. This simulation approach can help us understand the probability of success in a random maze environment, and also explore the impact of various factors such as maze complexity, size and starting position of the mouse on the outcome.
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Identify the measure of arc FE⏜ given the measure of arc FGC⏜ is 220∘
The value of the angle of the arc FE⏜ is calculated as: 20°
How to find the angle at the arc?The angle of an arc is identified by its two endpoints. The measure of an arc angle is found by dividing the arc length by the circle's circumference, then multiplying by 360 degrees. Formulas for calculating arcs and angles vary based on where they are in reference to the circle.
Now, from the given image, we see that:
FGC⏜ = 220°
∠B = 30°
∠OEC = ∠OCE = 30°
CDE⏜ = 220°
Thus:
FE⏜ = 360° - 220° - 120°
FE⏜ = 20°
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Jovi's garden is 32 feet long and 8 feet wide. He divides the garden into square sections that are 4 feet long then he plants a different type of vegetable in each section. What is the greatest number of types of vegetables that jovi can play in the garden
Answer:
Step-by-step explanation:
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In 1990, Jane became a real estate agent. Eight years later, she sold a house for $144,000. Eleven years later, she sold the same house for $245,000. Write an equation that represents the value of the house (V) related to the number of years (t) since Jane became a real estate agent. (Hint: Be careful! The second sale of the house was 11 years after the first sale which was 8 years after she became a real estate agent! That means the second sale took place 19 years after she became an agent!)
Let's break down the information given in the problem:
- Jane became a real estate agent in 1990.
- She sold a house 8 years later (in 1998) for $144,000.
- She sold the same house 11 years after that sale (in 2009), which is 19 years after she became an agent, for $245,000.
To write an equation that represents the value of the house (V) related to the number of years (t) since Jane became a real estate agent, we can use the information from the two sales to find the rate of change in the value of the house over time. We can use this rate of change to write an equation in point-slope form:
V - V1 = m(t - t1)
where V1 is the value of the house at time t1, m is the rate of change in the value of the house, and t is the time since Jane became a real estate agent.
Using the two sales, we can find the rate of change in the value of the house as follows:
m = (V2 - V1) / (t2 - t1)
where V2 is the value of the house at the second sale, t2 is the time of the second sale (19 years after Jane became an agent), V1 is the value of the house at the first sale, and t1 is the time of the first sale (8 years after Jane became an agent).
Substituting the given values, we get:
m = ($245,000 - $144,000) / (19 - 8) = $10,100 per year
Now we can use the point-slope form equation to find the value of the house at any time t since Jane became a real estate agent. Let's choose 1990 as our initial time (t1), so V1 = $0:
V - 0 = $10,100 (t - 0)
Simplifying, we get:
V = $10,100t
Therefore, the equation that represents the value of the house (V) related to the number of years (t) since Jane became a real estate agent is V = $10,100t. Note that this equation assumes a constant rate of change in the value of the house over time, which may not be accurate in real life.
A 2 yard piece of copper wire costs $9. 72. What is the price per foot
the price per foot of the copper wire is $1.62.
What is the arithmetic operation?
The basic mathematical operations are addition, subtraction, multiplication, and division, which involve manipulating two or more quantities. They are essential to the study of numbers, including the order of operations, and are fundamental to other mathematical areas such as algebra, data management, and geometry. Understanding the rules of arithmetic operations is crucial for solving problems that involve these operations.
There are 3 feet in one yard, so 2 yards of copper wire is equal to 6 feet.
To find the price per foot, we need to divide the total price by the number of feet:
Price per foot = Total price ÷ Number of feet
Price per foot = $9.72 ÷ 6
Price per foot = $1.62
Therefore, the price per foot of the copper wire is $1.62.
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How to show (sin 7x)/sin x = 64(cos x)^6 - 80(cos x)^4 +24(cos x)^2 - 1 ?
Following shows (sin 7x)/sin x = 64(cos x)^6 - 80(cos x)^4 +24(cos x)^2 - 1:
12sin^3 x - 8sin^
To show that (sin 7x)/sin x is equal to 64(cos x)^6 - 80(cos x)^4 + 24(cos x)^2 - 1, we can use trigonometric identities and algebraic manipulation.
Let's start with the left-hand side of the equation:
(sin 7x)/sin x
Using the trigonometric identity for sin(A + B):
sin(A + B) = sin A cos B + cos A sin B
We can rewrite sin 7x as sin (6x + x):
sin (6x + x) = sin 6x cos x + cos 6x sin x
Now we can substitute sin 7x with sin 6x cos x + cos 6x sin x:
(sin 6x cos x + cos 6x sin x)/sin x
Next, we can simplify this expression by dividing both terms by sin x:
(sin 6x cos x)/sin x + (cos 6x sin x)/sin x
The sin x term cancels out, leaving us with:
sin 6x cos x + cos 6x
Now, we can use the double-angle identity for sin 2A:
sin 2A = 2sin A cos A
To rewrite sin 6x cos x, we can treat it as sin 2A with A = 3x:
sin 6x cos x = 2sin 3x cos 3x
Next, we can use the triple-angle identity for sin 3A:
sin 3A = 3sin A - 4sin^3 A
To rewrite sin 3x, we can treat it as sin A with A = x:
sin 3x = 3sin x - 4sin^3 x
Substituting this into our expression:
2sin 3x cos 3x = 2(3sin x - 4sin^3 x) cos 3x
Expanding further:
= 6sin x cos 3x - 8sin^3 x cos 3x
Now, we can use the double-angle identity for cos 2A:
cos 2A = cos^2 A - sin^2 A
To rewrite cos 3x, we can treat it as cos A with A = x:
cos 3x = cos^2 x - sin^2 x
Substituting this into our expression:
6sin x cos 3x - 8sin^3 x cos 3x = 6sin x (cos^2 x - sin^2 x) - 8sin^3 x (cos^2 x - sin^2 x)
Expanding further:
= 6sin x cos^2 x - 6sin x sin^2 x - 8sin^3 x cos^2 x + 8sin^3 x sin^2 x
Now, we can use the Pythagorean identity for sin^2 x + cos^2 x:
sin^2 x + cos^2 x = 1
To rewrite sin^2 x, we can subtract cos^2 x from both sides:
sin^2 x = 1 - cos^2 x
Substituting this back into our expression:
= 6sin x (cos^2 x - (1 - sin^2 x)) - 8sin^3 x cos^2 x + 8sin^3 x sin^2 x
= 6sin x (2sin^2 x) - 8sin^3 x cos^2 x + 8sin^3 x sin^2 x
= 12sin^3 x - 8sin^
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A rectangular playing field lies in the interior of an elliptical track that is 50 yards wide and 110 yards long. What is the width of of the rectangular playing field if the width is located 15 yards from either vertex?
The width of the rectangular playing field is approximately 50 yards if the width is located 15 yards from either vertex.
To solve the problem, we can draw a diagram and use the properties of ellipses.
First, we note that the major axis of the ellipse is 110 yards and the minor axis is 50 yards. We can find the distance between the two foci of the ellipse using the formula c^2 = a^2 - b^2, where c is the distance between the foci, and a and b are the lengths of the semi-major and semi-minor axes.
c^2 = 110^2 - 50^2
c^2 = 10800
c ≈ 104.0
Next, we draw the two foci of the ellipse and the rectangle as shown in the diagram below. We are given that the width of the rectangle is 30 yards (15 yards from either vertex). x be the length of the rectangle.
A B
+-------+-------+
/ \
/ \
/ \
C D
\ /
\ /
\ /
+-------+-------+
E F
We can see that the length of the rectangle is equal to the distance between points A and B, and the width of the rectangle is equal to the distance between points C and D. Using the Pythagorean theorem, we can find the length of the rectangle.
AB^2 = AE^2 + EB^2
AB^2 = (a/2)^2 + (c - b/2)^2
AB^2 = (55)^2 + (104 - 15)^2
AB^2 = 3025 + 7225
AB = sqrt(10250)
AB ≈ 101.2
Therefore, the length of the rectangle is approximately 101.2 yards.
To find the width of the rectangle, we can use the fact that the distance between points C and D is equal to twice the distance between the center of the ellipse and the minor axis. The center of the ellipse is the midpoint of the major axis, and the distance from the center to the minor axis is 25 yards.
CD = 2 * 25 = 50
Therefore, the width of the rectangle is approximately 50 yards.
In summary, the width of the rectangular playing field is approximately 50 yards if the width is located 15 yards from either vertex.
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Ben earns $12. 50 per hour and $6 for each delivery he makes. He wants to earn $168 in an 8-hour workday.
Part A) Which equation could you use to solve for the least number of deliveries he must make to reach his goal?
Part B) What is the least number of deliveries he must make to reach his goal?
Answer:
ben earns $9 per hour and $6 for each delivery he makes
Step-by-step explanation:
If KJ=10, find the length of arc HJ
Answer:
Step-by-step explanation:
The arc length formula is
[tex]L=\frac{\theta}{360}*2\pi r[/tex]
Theta is the angle that intersects arc HJ. That measure is 180-122 which is 58 degrees. We put that into the formula along with the radius measure of 10 to get:
[tex]L=\frac{58}{360}*2\pi (10)[/tex] which gives us, rounded to the nearest hundredth,
L = 10.12 units
+ = a) Find a parametrization for the curve of intersection of 9y2 + z2 = 1 and xyz = 1 such that it is defined for all t + b) Same for surfaces x=y_and x2 - y2 =z. c) Same for surfaces x2 + y2 + z =
a) Parametrization for the curve of intersection of 9y2 + z2 = 1 and xyz = 1 is:
x = 1/(z(sqrt((1 - z^2)/9)))
y = ±sqrt((1 - z^2)/9)
z = t
b)Parametrization for the curve of intersection of x=y_and x2 - y2 =z is
x = t
y = t
z = 0
c)Parametrization for the curve of intersection of x2 + y2 + z = is
x = r cos(t)
y = r sin(t)
z = 1 - r
a) To find the curve of intersection of the two surfaces [tex]9y^2 + z^2 = 1[/tex] and xyz = 1:
We can solve for one variable in terms of the others.
For example, we can solve for y in terms of z and x using the first equation:
[tex]9y^2 + z^2 = 1[/tex]
[tex]9y^2 = 1 - z^2[/tex]
[tex]y^2 = (1 - z^2)/9[/tex]
y = ±sqrt([tex](1 - z^2)/9)[/tex]
Substituting this into the second equation, we get:
x(sqrt((1 - [tex]z^2)/9))z = 1[/tex]
x = 1/(z(sqrt((1 - [tex]z^2)/9)))[/tex]
So, a parametrization for the curve of intersection is:
x = 1/(z(sqrt((1 - [tex]z^2)/9)))[/tex]
y = ±sqrt((1 - z^2)/9)
z = t
This is defined for all t except at z = ±1.
b) To find the curve of intersection of the two surfaces x = y and [tex]x^2 - y^2 = z:[/tex]
We can substitute x = y into the second equation:
[tex]x^2 - y^2 = z[/tex]
[tex]y^2 - y^2 = z[/tex]
z = 0
So the curve of intersection is just the x = y line.
A parametrization for this line is:
x = t
y = t
z = 0
c) To find a parametrization for the surface [tex]x^2 + y^2 + z = 1:[/tex]
We can use cylindrical coordinates:
x = r cos(t)
y = r sin(t)
z = 1 - r
where 0 ≤ r ≤ 1 and 0 ≤ t < 2π.
This parameterization covers the surface of a unit cylinder with its top and bottom caps removed.
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If 3 quarts is greater then 4 prints is that an equivalent measure
If 3 quarts is greater than 4prints, then the measure is not equivalent.
What is equivalent measurement?Equivalent units can be used to convert different units to the same unit for comparison. Equivalent means equal. For example , 1 kilogram is equal to 1,000 grams.
For example,
3 teaspoons = 1 tablespoon.
4 tablespoons = 1/4 cup.
5 tablespoons + 1 teaspoon = 1/3 cup.
8 tablespoons = 1/2 cup.
1 quart = 2pints
therefore 3 quarts = 2×3 = 6pints
therefore the statement that 3 quarter is greater than 4 prints is true and not an equivalent measure.
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What is question asking??? All I need is one example and I get the rest I just don’t understand the assignment
So basically you have to drag an angle to the box, example drag DFA to the box, then add what angle it is equal to. You have to calculate the angle and drag it opposite to the DFA. For DFA, it will be 58.
The tip of a pendulum swings through an arc whose length is 8 centimeters. If the radius of the pendulum is 26 centimeters, then what radian angle did the tip rotate through? Round your answer to the nearest hundredth
Answer: Therefore, the tip of the pendulum rotated through an angle of approximately 0.31 radians.
Step-by-step explanation:
The length of the arc traveled by the tip of the pendulum is 8 centimeters, and the radius of the pendulum is 26 centimeters. We can use the formula for the length of an arc of a circle to find the radian angle rotated through:
Length of arc = radius * angle in radians
Solving for the angle in radians, we get:
Angle in radians = Length of arc / radius
Plugging in the given values, we get:
Angle in radians = 8 / 26
Simplifying this expression, we get:
Angle in radians = 0.30769...
Rounding this value to the nearest hundredth, we get:
Angle in radians = 0.31
Therefore, the tip of the pendulum rotated through an angle of approximately 0.31 radians.
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1. bona drives tor 3 hours at 44mph. clare drives 144 mies in 4 hours. how
for would bena travel if she drove for 3 hours at the same speed os
claire
2. janet and andrew leave their home at the same time. janet has 60
milles to travel and drives at 40 mph. andrew have 80 miles to travel
and also drives at 40 mph
a) how long does janets journey take?
(b) how much longer does andrew spend driving than janeta
1) Bona can drive 108 miles if she drove for 3 hour at the same speed as Claire.
2) Janet would take 1.5 hour to complete the journey and Andrew spend half hour more driving than Janet.
1) Bona speed is 44mph
Time taken by Bona is 3 hour
distance travelled by Claire is 144 miles
Time taken by Claire is 4 hour
Claire's speed = distance / time
Claire's speed = 144/4
Claire's speed = 36 mph
Distance travelled by Bona = speed × time
Distance travelled by Bona = 36 × 3
Distance travelled by Bona = 108 miles
2) Janet distance = 60 miles
Janet speed = 40 mph
Time taken by Janet = distance / speed
Time taken by Janet = 60/40
Time taken by Janet = 1.5 hour
Janet would take 1.5 hour to complete the journey
Andrew distance = 80 miles
Andrew speed = 40 mph
Time taken by Andrew = distance / speed
Time taken by Andrew = 80/40
Time taken by Andrew= 2 hour
Andrew spend half hour more driving than Janet.
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Using technology, calculate the weighted mean of the RORs for each portfolio. Based on the results, which list shows a comparison of the overall performance of the portfolios, from best to worst?
The ranking from worst to best is Portfolio 3, Portfolio 2, and Portfolio 1. So, correct option is B.
To calculate the weighted mean of the RORs for each portfolio, we need to first multiply each ROR by the corresponding portfolio value and then sum the products for each portfolio. We then divide the total by the sum of the portfolio values.
The weighted mean for Portfolio 1 = [(10.4% x $700) + (-29.7% x $12,000) + (37.2% x $600) + (7.5% x $4,400) + (6.3% x $250)] / ($700 + $12,000 + $600 + $4,400 + $250) = -16.8%
Similarly, the weighted mean for Portfolio 2 = 3.8% and for Portfolio 3 = 11.2%.
Based on the results, the list that shows a comparison of the overall performance of the portfolios from worst to best is option b) Portfolio 3, Portfolio 2, Portfolio 1. Portfolio 3 has the highest weighted mean return of 11.2%, followed by Portfolio 2 with a return of 3.8%, and Portfolio 1 has the lowest weighted mean return of -16.8%.
Therefore, correct option is B.
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Complete question is:
ROR Portfolio 1 Portfolio 2 Portfolio 3
10.4% $700 $6,000 $3,500
-29.7% $12,000 $9,000 $5,500
37.2% $600 $4,500 $5,750
7.5% $4,400 $2,000 $1,500
6.3% $250 $1,100 $4,500
Calculate the weighted mean of the RORs for each portfolio. Based on the results, which list shows a comparison of the overall performance of the portfolios, from worst to best?
a) Portfolio 3, Portfolio 1, Portfolio 2
b) Portfolio 3, Portfolio 2, Portfolio 1
c) Portfolio 1, Portfolio 2, Portfolio 3
d) Portfolio 1, Portfolio 3, Portfolio 2
given m||n, find the value of x
Answer:
Step-by-step explanation:
The value of x is 19 if the lines l and m are parallel.
The lines l and m are parallel
The corresponding angles are equal
We have to find the value of x
6x-9 = 5x+10
Let us take all the variable terms on one side
6x-5x=10+9
x=19
Hence, the value of x is 19 if the lines l and m are parallel.
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Need help with these questions!! Person who answers will be marked brainliest
Answer:
1.5 for both
Step-by-step explanation:
Taking each number and multiplying it by 1.5 will get you the dilated coordinates
suppose x is a random variable with mean mu and standard deviation sigma. If a large number of trials are observed, at least what percentage of these values is expected to lie between mu minus 2 sigma and mu plus 2 sigma?
At least 95% of the observed values are expected to lie between mu minus 2 sigma and mu plus 2 sigma.
This is because of the empirical rule, also known as the 68-95-99.7 rule, which states that in a normal distribution, approximately 68% of the observations will fall within one standard deviation of the mean, about 95% of the observations will fall within two standard deviations of the mean, and around 99.7% of the observations will fall within three standard deviations of the mean.
In this case, we are given that x has mean mu and standard deviation sigma. Therefore, about 95% of the values of x are expected to lie between mu minus 2 sigma and mu plus 2 sigma, as this interval covers two standard deviations on either side of the mean.
Mathematically, we can express this as:
P(mu - 2sigma < x < mu + 2sigma) ≈ 0.95
where P is the probability that x falls within the interval mu - 2sigma to mu + 2sigma.
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if I draw a marble 48 times a white marble is selected 35 times ana a yellow one is selected 13 times what is the probability of the next one to be yellow
A 13%
B 27%
C 51%
D 63%
If we are drawing without replacement, the probability is approximately 27.7%, which is closest to option B: 27%.
What is the probability that the next marble is yellow?The probability of drawing a yellow marble on the next draw depends on whether we are drawing with or without replacement.
If we are drawing without replacement, then the probability of drawing a yellow marble is 13 out of the remaining 47 marbles, since we have already drawn 35 white marbles and 13 yellow marbles out of the 48 total marbles.
If we are drawing with replacement, then the probability of drawing a yellow marble on the next draw is still 1/3, or approximately 33.3%.
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what is 88 closer to 64 or 125
Answer:
64
Step-by-step explanation:
Answer:
64...i think
Step-by-step explanation:
125 - 88=37
88-64=24
PLS MARK BRAINLIEST
The given segment is the diameter of a circle bar cd the coordinates of c are (-3,5) and the coordinates of d are (6,-2) . find the center of the circle
To find the center of the circle, we need to find the midpoint of the diameter segment CD.
Using the midpoint formula, we can find the coordinates of the midpoint M:
Midpoint formula:
M = ( (x1 + x2)/2 , (y1 + y2)/2 )
Plugging in the coordinates of C (-3,5) and D (6,-2):
M = ( (-3 + 6)/2 , (5 - 2)/2 )
M = (1.5, 1.5)
Therefore, the center of the circle is at point M with coordinates (1.5, 1.5).
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If f(x) = x2 − 6x − 4 and g(x) = 5x + 3, what is (f + g)(−3)? (1 point)
41
35
11
−35
As per the given information in the question, we can compute that the correct option is C) 11.
According to the question:
[tex]f(x)=x^{2} -6x-4[/tex]
[tex]g(x)=5x+3[/tex]
Therefore,
[tex](f+g)(x)= f(x)+g(x) \\
=x^{2} -6x-4+5x+3 \\
=x^2-x-1[/tex]
Now, to find (f+g)(-3):
substituting x=-3 in the above equation
we get,
[tex](f+g)(-3)= (-3)^2-(-3)-1 \\
= 9+3-1 \\
=11[/tex]
Hence (f+g)(-3)=11
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Julia just lit a new candle and then let it burn all the way down to nothing. The candle burned at a rate of 0.75 inches per hour and its initial length was 9 inches. Write an equation for
L
,
L, in terms of
t
,
t, representing the length of the candle remaining unburned, in inches,
t
t hours after the candle was lit.
The required equation for the length of the candle is L = 9-0.75t.
Given that a 9-inch candle burns at the rate of 0.75 in per hour we need to determine the equation for the length L of the candle when it burns for t hours,
So, if the candle is burning at the rate of 0.75 in per hour so after burning for t hour the length decreases by 0.75t, from the original length i.e 9 in
So, we can write the equation as =
L = 9-0.75t
Hence the required equation for the length of the candle is L = 9-0.75t.
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Use the differential dz to approximate the change that will be
observed in z = f (x, y) = 5/x^2 + y^2 as x changes from −1 to
−0.93 and y changes from 2 to 1.94.
To approximate the change in z = f(x, y) as x changes from −1 to −0.93 and y changes from 2 to 1.94, we can use the differential dz.
First, we need to find the partial derivatives of f with respect to x and y:
∂f/∂x = -10/x³(y²)
∂f/∂y = -10(x²)/y³
Then, we can use the following formula:
dz ≈ ∂f/∂x * Δx + ∂f/∂y * Δy
where Δx and Δy are the changes in x and y, respectively.
Substituting in the given values, we have:
Δx = -0.93 - (-1) = 0.07
Δy = 1.94 - 2 = -0.06
Using the partial derivatives we calculated earlier, we get:
dz ≈ (-10/-1.037³(2²)) * 0.07 + (-10((-1)²)/1.94³) * (-0.06)
dz ≈ -0.031
Therefore, the approximate change observed in z as x changes from −1 to −0.93 and y changes from 2 to 1.94 is -0.031.
To approximate the change in z using the differential dz, we first need to find the partial derivatives of z with respect to x and y. Given z = f(x, y) = 5/(x² + y²):
∂z/∂x = -10x/(x² + y²)²
∂z/∂y = -10y/(x² + y²)²
Now, we need to find the differential dz:
dz = (∂z/∂x)dx + (∂z/∂y)dy
Since x changes from -1 to -0.93, dx = -0.93 - (-1) = 0.07. Similarly, y changes from 2 to 1.94, so dy = 1.94 - 2 = -0.06.
Now, plug in the initial values of x and y (-1, 2):
∂z/∂x = -10(-1)/((-1)² + 2²)² = -10/25
∂z/∂y = -10(2)/((-1)² + 2²)² = -40/25
Now, plug in dx and dy into the dz equation:
dz = (-10/25)(0.07) + (-40/25)(-0.06) = 0.28 - 0.096 = 0.184
So, the approximate change in z when x changes from -1 to -0.93 and y changes from 2 to 1.94 is 0.184.
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3.
What does limit as x goes to infinity of the quotient of f of x and g of x equals 5 show? (4 points)
g(x) grows faster than f(x) as x goes to infinity.
f(x) and g(x) grow at the same rate as x goes to infinity.
f(x) grows faster than g(x) as x goes to infinity.
L'Hôpital's Rule must be used to determine the true limit value
The statement "limit as x goes to infinity of the quotient of f of x and g of x equals 5" means that as x gets larger and larger, the ratio of f(x) to g(x) approaches 5.
If f(x) grows faster than g(x) as x goes to infinity, then the ratio of f(x) to g(x) would approach infinity, not 5.
If f(x) and g(x) grow at the same rate as x goes to infinity, then the ratio of f(x) to g(x) would approach a constant value, not necessarily 5.
Therefore, the statement "limit as x goes to infinity of the quotient of f of x and g of x equals 5" shows that g(x) grows faster than f(x) as x goes to infinity. L'Hôpital's Rule is not necessary to determine this conclusion.
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The seventh-grade students at Charleston Middle School are choosing one girl and one boy for student council. Their choices for girls are Michaela (M), Candice (C), and Raven (R), and for boys, Neil (N), Barney (B), and Ted (T). The sample space for the combined selection is represented in the table. Complete the table and the sentence beneath it.
Boys
Neil Barney Ted
Girls Michaela N-M
B-M
T-M
Candice N-C
B-C
T-C
Raven N-R
B-R
T-R
If instead of three girls and three boys, there were four girls and four boys to choose from, the new sample size would be ?
If there were four girls and four boys to choose from, the new sample size is one that will be a total of 16 possible combinations.
What is the sample about?When making selections for the student council's representatives, the sample size belongs to the whole of the feasible options.
To know the sample size, we must reason every conceivable permutation involving a single female and a single male.
So, if there were four girls and four boys to choose from, the new sample size would be:
There are:
4 choices for girls
4 choices for boys.
So, for each girl's choice, there would be 4 corresponding choices for boys.
Hence it will be:
4 x 4 = 16
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Madison swims 2/5 mile in an hour. Declan swims 7/10 mine in 1/5 hour. Who swims faster?
Declan swims faster than Madison, with a speed of 7/2 miles per hour compared to Madison's speed of 2/5 miles per hour.
How to compare swimming speed?Declan swims faster than Madison. While Madison swims 2/5 mile in an hour, which means her speed is 2/5 miles per hour, Declan swims 7/10 mile in 1/5 hour, which means his speed is 7/2 miles per hour. This indicates that Declan's speed is greater than Madison's speed.
In fact, we can see that Declan's speed is almost 4 times faster than Madison's speed. This means that Declan can cover a greater distance in the same amount of time compared to Madison.
Therefore, based on the given information, we can confidently say that Declan swims faster than Madison.
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