a) The radius of convergence is also 1, since it is the same as the radius of convergence of ln(1+x).
a) We can find a power series representation for f(x) = ln(1+x) by using the formula:
ln(1 + x) = ∑(-1)^(n-1) * x^n / n for |x| < 1
So, the power series representation for ln(1+x) is:
ln(1 + x) = x - x^2/2 + x^3/3 - x^4/4 + x^5/5 - ...
The radius of convergence is 1, since the series converges for |x| < 1 and diverges for |x| > 1.
b) To find a power series for f(x) = xln(1+x), we can use the product rule of power series:
xln(1 + x) = x * [x - x^2/2 + x^3/3 - x^4/4 + x^5/5 - ...]
= x^2 - x^3/2 + x^4/3 - x^5/4 + x^6/5 - ...
So, the power series representation for f(x) = xln(1+x) is:
f(x) = xln(1+x) = ∑(-1)^(n-1) * x^(n+1) / n for |x| < 1
c) To find a power series for f(x)ln(x^2 + 1), we can use the product rule of power series again:
f(x)ln(x^2 + 1) = [x - x^2/2 + x^3/3 - x^4/4 + x^5/5 - ...] * ln(x^2 + 1)
= [xln(x^2 + 1)] - [x^2 ln(x^2 + 1)]/2 + [x^3 ln(x^2 + 1)]/3 - [x^4 ln(x^2 + 1)]/4 + [x^5 ln(x^2 + 1)]/5 - ...
We already have a power series for xln(1+x), which is the same as xln(x^2+1) for |x| < 1. So, we can substitute it in the above series:
f(x)ln(x^2 + 1) = ∑(-1)^(n-1) * x^(2n+1) / (2n+1) * [1 - 1/2 + 1/3 - 1/4 + 1/5 - ...]
= ∑(-1)^(n-1) * x^(2n+1) / (2n+1) * ln(x^2 + 1)
The radius of convergence of this power series is also 1, since it is the same as the radius of convergence of ln(1+x) and xln(1+x).
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A grocery shop sells apples and pears.
5 apples cost 60p.
8 pears cost 72p.
a) Work out the cost of 1 apple, in pence (p).
b) Work out the cost of 1 pear, in pence (p).
c) Which type of fruit is cheaper per item?
Answer:
pear
Step-by-step explanation:
60/5 = 12
72/8 = 9
so one pear is cheaper than one apple
Dilations on coordinate plane
The Rule for the dilation is (x, y) → (2x, 2y).
The rule that best represents the dilation applied to Triangle E to create Triangle F is:
(x, y) → (2x, 2y)
This rule shows that the dilation is centered at the origin and has a scale factor of 2, which means that all the coordinates of Triangle E are multiplied by 2 to obtain the corresponding coordinates of Triangle F.
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Tom needs to paint a fence that is made of 5 panels. He has red, yellow, green, blue, and white paint. In how many ways can Tom paint the fence if no two neighboring panels can be painted the same color?
Answer: In how many ways can Tom paint the fence if no two neighboring panels can be painted the same color?
Step-by-step explanation:
There are a total of 120 ways to paint the fence with no two neighboring panels painted the same color.
To see why, consider the first panel. Tom can paint it any of the 5 colors. Without loss of generality, assume that he paints it red.
For the second panel, he can choose any of the 4 remaining colors (since he cannot use red again). Without loss of generality, assume that he paints it yellow.
For the third panel, he can choose any of the 3 remaining colors (since he cannot use red or yellow again). Without loss of generality, assume that he paints it green.
For the fourth panel, he can choose any of the 2 remaining colors (since he cannot use red, yellow, or green again). Without loss of generality, assume that he paints it blue.
For the fifth panel, he can choose the only remaining color (since he cannot use red, yellow, green, or blue again). Without loss of generality, assume that he paints it white.
Therefore, there are 5 × 4 × 3 × 2 × 1 = 120 ways to paint the fence with no two neighboring panels painted the same color.
{Hope This Helps! :)}
What is the relationship between the two radios 10/24 and 5/12
Answer: These two ratios, 10/24 and 5/12 share a proportional relationship.
Step-by-step explanation:
Because when you divide the denominator and the numerator (10/24) by two they’ll equal the other ratio.
A softball pitcher has a 0.431 probability of throwing a strike for each pitch. If the softball pitcher throws 22 pitches, what is the probability that exactly 12 of them are strikes?
Round your answer to 2 decimal places.
The probability that exactly 12 out of 22 pitches are strikes is approximately 0.18 (rounded to two decimal places).
To find the probability that exactly 12 out of 22 pitches are strikes, we can use the binomial probability formula. In this case, the probability of throwing a strike for each pitch is 0.431, and we want to calculate the probability of getting exactly 12 strikes out of 22 pitches.
Using the binomial probability formula, the probability of getting exactly 12 strikes out of 22 pitches is:
P(X = 12) = C(22, 12) * (0.431)^12 * (1 - 0.431)^(22 - 12)
where C(22, 12) represents the number of ways to choose 12 strikes out of 22 pitches.
Calculating this probability, we find:
P(X = 12) = 22! / (12! * (22 - 12)!) * (0.431)^12 * (1 - 0.431)^(22 - 12)
P(X = 12) ≈ 0.1832
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suppose we have 3 numbers, none of which are equal. suppose we know the mean of the 3 numbers is 20, the median is 18, and the range (largest-smallest) is 10. what are the 3 numbers?
To find the three numbers, we need to use the information given about their mean, median, and range. Since the median is 18, we know that the middle number of the three must be 18. Also, since the mean is 20, the sum of the three numbers must be 3 times 20, which is 60. So, the 3 numbers are 16, 18, and 26.
Let's call the smallest number x, then the largest number must be x+10 since the range is 10.
Now we can set up an equation to solve for x. We know that the sum of the three numbers is 60, so we can write:
x + 18 + (x+10) = 60
Simplifying this equation gives:
2x + 28 = 60
Subtracting 28 from both sides gives:
2x = 32
Dividing by 2 gives:
x = 16
Therefore, the three numbers are 16, 18, and 26.
Given the information provided, we know the following:
1. Mean of 3 numbers is 20.
2. Median of the 3 numbers is 18.
3. Range (largest - smallest) is 10.
Since there are 3 numbers and the median is 18, the middle number is 18. Let's call the smallest number "x" and the largest number "y". We can now write two equations:
1. (x + 18 + y) / 3 = 20
2. y - x = 10
We can simplify the first equation to find the sum of the 3 numbers:
x + 18 + y = 60
Now, we can solve for x in the second equation:
x = y - 10
Substituting the value of x in the first equation:
(y - 10) + 18 + y = 60
Solving for y:
2y = 52
y = 26
Now, we can find the value of x:
x = 26 - 10
x = 16
So, the 3 numbers are 16, 18, and 26.
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100 POINTS IF YOU DO THIS CORRECT
This stem-and-leaf plot represents the height of the students on Mark’s basketball team. One student’s height is missing from the plot. If the mean height of all the students on the team is 63 inches, what is the unknown height?
in.____
The unknown height of the student on the team, given the stem and leaf plot, would be 60 inches
How to find the unknown height ?To find the unknown height, we first have to list the heights of the known students on the team. From the stem and leaf plot, the heights are :
57, 59, 62, 65, 68, 70
The mean height of all the students is 63 inches which means that if this mean is denoted as x, the unknown height would be:
63 = ( 57 + 59 + 62 + 65 + 68 + 70 + x ) / 7
Solving for x gives:
7 x 63 = ( 381 + x )
441 = 381 + x
x = 441 - 381
x = 60 inches
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9. Write an equation of an ellipse in standard form with the center at the origin and a height of 3 units and width of 1 unit.
The equation for the ellipse in standard form with the center at the origin and a height of 3 units and width of 1 unit is given as follows:
x²/0.25 + y²/2.25 = 1
How to obtain the equation of the ellipse?Considering the center at the origin, the format for the equation of the ellipse is given as follows:
x²/a² + y²/b² = 1.
The ellipse has a height of 3 units, hence the parameter b is given as follows:
2b = 3
b = 1.5.
Hence the square is of:
b² = 1.5² = 2.25.
The ellipse has a width of 1 unit, hence the parameter a is given as follows:
2a = 1
a = 0.5.
Then the square is of:
a² = 0.5² = 0.25.
Then the equation is given as follows:
x²/0.25 + y²/2.25 = 1
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Jenny changed £480 into $744 when she went to America.
On her return she had $310 left.
If the exchange rate stayed the same, how many pounds did she get back?
Answer:
£200
Step-by-step explanation:
To determine how many pounds Jenny got back, we first need to calculate the exchange rate between pounds (£) and dollars ($).
Jenny initially changed £480 into $744, so £480 = $744.
To find the exchange rate in terms of pounds (£) to dollars ($), divide both quantities by 480:
[tex]\dfrac{480}{480}=\dfrac{744}{480}[/tex]
[tex]\£1=\$1.55[/tex]
Therefore, the exchange rate was £1 = $1.55.
This means that for every £1 Jenny exchanged, she received $1.55.
To convert the $310 Jenny had left back into pounds (£), divide the dollar amount by 1.55:
[tex]\dfrac{310}{1.55}=200[/tex]
Therefore, if the exchange rate stayed the same, Jenny got back £200.
it's a good deal bro take it
Answer:
3/4
Step-by-step explanation:
9/12
divide top and bottom by 3
3/4
Convert the integral below to polar coordinates and evaluate the integral. Integral 5/root 2 0 Integral root 25 - y^2 y xy dx dy Instructions: Please enter the integrand in the first answer box, typing theta for theta. Depending on the order of integration you choose, enter dr and d theta in either order into the second and third answer boxes with only one dr or d theta in each box. Then, enter the limits of integration and evaluate the integral to find the volume. Integral B A Integral D C A = B = C = D = Volume =
The given integral is: ∫∫R 5/√(2) xy dA, where R is the region in the xy-plane bounded by the curves y = 0, x = √(25 - y^2) and x = 0. The volume of the solid is (5^5/8) cubic units.
Converting to polar coordinates, we have:
x = r cos(θ), y = r sin(θ), and the limits of integration become:
0 ≤ θ ≤ π/2, 0 ≤ r ≤ 5.
Also, the differential of area becomes dA = r dr dθ.
Substituting for x and y, and dA, we have:
∫∫R 5/√(2) xy dA = ∫θ=0π/2 ∫r=0^5 5/√(2) (r cos(θ)) (r sin(θ)) r dr dθ
= 5/√(2) ∫θ=0π/2 ∫r=0^5 r^3 cos(θ) sin(θ) dr dθ
= 5/√(2) ∫θ=0π/2 [sin(θ)/4] ∫r=0^5 r^4 cos(θ) dr dθ
= 5/√(2) ∫θ=0π/2 [sin(θ)/4] [(5^5 cos(θ))/5] dθ
= (5^5/4√(2)) ∫θ=0π/2 sin(θ) cos(θ) dθ
= (5^5/8)
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Two cars drive from one end to the other, in opposite directions. Each goes at a constant speed. One car takes 30 seconds to get from one end to the other. The other takes 50 seconds. They both reach the end at the same time. Draw a graph and use it to find how far the cars are from each end of the track when they pass one another.
The first car is at a Distance 1/80 units away from the starting end of the track when they pass. The second car is 79/80 units away from the starting end of the track when they pass.
Let's represent the distance traveled by the first car as a function of time, and the distance traveled by the second car as another function of time. We'll assume that the track is of length 1 unit (for simplicity).
The first car's distance as a function of time can be represented by the equation:
d1(t) = t/30
The second car's distance as a function of time can be represented by the equation:
d2(t) = 1 - t/50
To find when the cars pass each other, we need to find the time at which their distances are equal. So we set d1(t) = d2(t) and solve for t:
t/30 = 1 - t/50
Simplifying the equation:
50t = 30 - 30t
80t = 30
t = 30/80 = 3/8
Therefore, the cars pass each other at t = 3/8 seconds. To find the distance from each end of the track when they pass, we substitute this time value into either d1(t) or d2(t):
d1(3/8) = (3/8)/30 = 1/80
The first car is 1/80 units away from the starting end of the track when they pass. Since the track is of length 1 unit, the second car is 1 - 1/80 = 79/80 units away from the starting end of the track when they pass.
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A square of area 36 cm² is cut to make two rectangles, A and B.
Area = 36 cm²
The ratio of area A to area B is 2:1
Work out the dimensions of rectangles A and B.
Rectangle A
Length:
Width:
Rectangle B
Length:
Width:
Based on the information, we can infer that the dimensions of rectangle A are length = 2√6 cm and width = 3√2 cm, and the dimensions of rectangle B are length = 4√6 cm and width = √6 cm.
How to find the dimensions of the rectangle?To find the dimensions of the rectangle we must take into account the dimensions of the square model as a reference to know the dimensions of the rectangle. Let's represent the dimensions of rectangle A as x and y, and the dimensions of rectangle B as z and w.
We know that the area of rectangle A plus the area of rectangle B equals 36 cm²:
xy + zw = 36We also know that the ratio of the area of rectangle A to rectangle B is 2:1:
xy : zw = 2:1We can use this ratio to write one of the dimensions of rectangle B in terms of one of the dimensions of rectangle A:
zw = 1/2xySubstituting this expression into the first equation, we get:
xy + 1/2xy = 36Simplifying:
3/2xy = 36xy = 24Now we can use the ratio again to find the values of z and w:
z/w = 2/1z = 2wSubstituting this into the area equation for rectangle B:
2w*w = zw = 1/2xy = 12Solving for w:
2w² = 12w² = 6w = √6Finally, we can find the values of x and y using the fact that xy = 24:
y = 24/xSubstituting into the ratio expression for rectangle B:
z/w = 2/12w/w = 2/1z = 2w = 2√6x(24/x) = 24 = yxSolving for x:
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HELP ME PLEASE
A composite figure is shown.
A five-sided figure with two parallel sides. The shorter one is 18 meters. The height of the figure is 12 meters. The portion from the vertex to the perpendicular height is 7 meters. The portion from a point to a vertical line created by two vertices is 5 meters.
Which of the following represents the total area of the figure?
288 m2
360 m2
416 m2
576 m2
Ram Purchased a flat at ₹1. 1 lakh and Prem purchased aplot of land worth ₹ 1. 1 lakh. The respective annual rates at which the prices of the flat and the plot increases were 10% and 5%. After two years they exchanged their belongings and one paid the other the difference. Then who paid to whom by how much?
Prem purchased a plot for ₹1.1 lakh and its price increased at an annual rate of 5%.Ram's flat had a higher value than Prem's plot, Prem paid Ram the difference, which was ₹0.118 lakh or ₹11,800. Ram's flat is more valuable than Prem's plot of land
After two years, the value of Ram's flat would be ₹1.1 lakh + (10% of ₹1.1 lakh × 2 years) = ₹1.43 lakh.
Similarly, the value of Prem's plot of land would be ₹1.1 lakh + (5% of ₹1.1 lakh × 2 years) = ₹1.21 lakh.
Therefore, the difference in value between Ram's flat and Prem's plot of land is ₹1.43 lakh - ₹1.21 lakh = ₹22,000.
Ram would have to pay ₹22,000 to Prem as Ram's flat is more valuable than Prem's plot of land.
Ram purchased a flat for ₹1.1 lakh and its price increased at an annual rate of 10%. After two years, the flat's value would be:
1.1 lakh * (1 + 0.1)^2 = 1.1 lakh * 1.21 = ₹1.331 lakh
Prem purchased a plot for ₹1.1 lakh and its price increased at an annual rate of 5%. After two years, the plot's value would be:
1.1 lakh * (1 + 0.05)^2 = 1.1 lakh * 1.1025 = ₹1.213 lakh
After exchanging their properties, the difference in value is:
₹1.331 lakh (flat) - ₹1.213 lakh (plot) = ₹0.118 lakh
Since Ram's flat had a higher value than Prem's plot, Prem paid Ram the difference, which was ₹0.118 lakh or ₹11,800.
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The table shows the number of laps Candice and her two friends ran each day for five days. Drag the names to order them from the least consistent number of laps each day (at the bottom) to the most consistent number of laps each day (at the top).
Candice ran the most consistent number of laps each day based on the mean absolute deviation.
let's calculate the mean number of laps for each friend:
Candice: (5 + 6 + 8 + 5 + 7) / 5 = 31 / 5 = 6.2 laps
Malaya: (4 + 5 + 3 + 3 + 5) / 5 = 20 / 5 = 4 laps
Zoe: (7 + 8 + 6 + 8 + 8) / 5 = 37 / 5 = 7.4 laps
Next, we calculate the absolute deviation of each data point from the mean for each friend:
Candice: |5 - 6.2| = 1.2, |6 - 6.2| = 0.2, |8 - 6.2| = 1.8, |5 - 6.2| = 1.2, |7 - 6.2| = 0.8
Malaya: |4 - 4| = 0, |5 - 4| = 1, |3 - 4| = 1, |3 - 4| = 1, |5 - 4| = 1
Zoe: |7 - 7.4| = 0.4, |8 - 7.4| = 0.6, |6 - 7.4| = 1.4, |8 - 7.4| = 0.6, |8 - 7.4| = 0.6
Now, we calculate the MAD for each friend by taking the average of the absolute deviations:
Candice: (1.2 + 0.2 + 1.8 + 1.2 + 0.8) / 5 = 0.6 laps
Malaya: (0 + 1 + 1 + 1 + 1) / 5 = 0.8 laps
Zoe: (0.4 + 0.6 + 1.4 + 0.6 + 0.6) / 5 = 0.72 laps
Comparing the MAD values, we see that Candice has the lowest MAD (0.6 laps), followed by Zoe (0.72 laps), and Malaya (0.8 laps).
Therefore, Candice ran the most consistent number of laps each day based on the mean absolute deviation.
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The table shows the number of laps Candice and her two friends ran each day for five days. which friend ran the most consistent number of laps each day? use the mean absolute deviation to construct an argument to justify the answer
Girl Day 1 Day 2 Day 3 Day 4 Day 5
Candice 5 6 8 5 7
Malaya 4 5 3 3 5
Zoe 7 8 6 8 8
assumptions:all employees in department a attend the sales training employees in department a attend the technical training employees who attend both training programmes are in department b.conclusion:all employees who attend the sales training programme and who attend the technical training programme are in department the assumptions are true, is the conclusion:
The conclusion that "all employees who attend the sales training program and who attend the technical training program are in department B" does not necessarily follow from the given assumptions.
The assumptions state that all employees in department A attend the sales training and employees in department A attend the technical training. It also mentions that employees who attend both training programs are in department B. However, it does not provide any information about employees who only attend one of the training programs.
Based on the given information, we cannot conclude that all employees who attend both training programs are in department B. There could be employees from department A who attend both programs and employees from department B who only attend one of the programs.
Therefore, the conclusion cannot be determined solely based on the given assumptions. Additional information about the distribution of employees across departments and their training program attendance is needed to draw a conclusive statement about the department affiliation of employees attending both training programs.
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copper has a face-centered cubic unit cell. how many atoms of cu are present in each unit cell?
The total number of copper (Cu) atoms in each face-centered cubic (FCC) unit cell is 7 atoms.
Here, we have,
In a face-centered cubic (FCC) unit cell, there are a total of 4 atoms.
Each corner of the unit cell contains 1/8th of an atom,
and since there are 8 corners in a cubic unit cell,
the total contribution from the corners is 8 * 1/8 = 1 atom.
Additionally, there is 1 atom located at the center of each face,
and since there are 6 faces in a cubic unit cell,
the contribution from the faces is 6 * 1 = 6 atoms.
Therefore, the total number of copper (Cu) atoms in each face-centered cubic (FCC) unit cell is 1 + 6 = 7 atoms.
Hence, 7 atoms of cu are present in each unit cell.
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Find the height of the figure with the volume V=480pi cubic cm
The value of the height is 10/3 cm
How to determine the heightFirst, we need to know that the formula for calculating the volume of a cylinder is expressed as;
V = πr²h
Such that the parameters are expressed as;
V is the volume of the cylinder.r is the radius of the cylinderh is the height of the cylinderπ takes the constant value of 3.14Now, substitute the values, we have that;
r = 12cm
Volume = 480π cm³
Then,
480π = 144π × h
divide both sides by the coefficient of h, we get;
h = 480π/144π
h = 40/12
h = 20/6
h = 10/3 cm
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Find x to the nearest degree
Answer:
56°
Step-by-step explanation:
call the unknown side length L.
then 5² + L² = 9²
L² = 9² - 5²
=81 - 25
= 56
L =√56
(sin x)/√56 = (sin 90)/9
sin x = (√56 X sin 90)/9 = √56 /9
x = sin^-1 (√56 / 9)
= 56.25°
= 56° to nearest degree
What is the area of ΔABC such that b = 28 cm, c = 14 cm, and m∠A = 30°? Round the answer to three decimal places.
98 cm2
169.741 cm2
193.654 cm2
196 cm2
The Area of ΔABC is 98 cm2.
The area of the triangle, we can use the formula:
A = 0.5 * b * c * sin(A)
where A is the angle between sides b and c.
Given that b = 28 cm, c = 14 cm, and m∠A = 30°, we can calculate sin(A) as follows
sin(A) = sin(30°) = 0.5
Substituting the values in the formula, we get:
A = 0.5 * 28 * 14 * 0.5 = 98 cm2
Therefore, the area of ΔABC is 98 cm2.
Hence, the correct option is: 98 cm2.
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Which of the following is a function? y = |x| + 3 y = x2 + 3x y = 2x + 1 y = x2 - 5x
A function is a relation between a set of inputs (called the domain) and a set of outputs (called the range) where each input is associated with exactly one output.
Out of the given options, the function is:
y = 2x + 1
In this equation, for every value of x, there is a unique corresponding value of y. Therefore, it satisfies the definition of a function.
The other options:
y = |x| + 3: This equation represents the absolute value of x, which means that for negative values of x, the output will be the positive counterpart. Thus, it does not satisfy the definition of a function as it does not have a unique output for every input. y = x^2 + 3x: This equation represents a quadratic function. It does satisfy the definition of a function, as each input has a unique output.y = x^2 - 5x: This equation also represents a quadratic function and satisfies the definition of a function.[tex][/tex]find the product of mode and median of the set:8,3,1,7,3,4,8,3,4,9,5
Answer:
First we put the number in order
(1 3 3 3 4 4 5 7 8 8 9)
The mode is the number that appears the most.
from the above order 3 is repeated more so 3 is the mode
The median is the middle value of a set of numbers.
from (1 3 3 3 4 4 5 7 8 8 9) the middle value is 4 so the median is 4
⇒Then the product of the mode and median
3*4=12
Write a paragraph about the transformations you used to create your logo.
You must use at least:
-1 reflection
-1 translation
-1 rotation
-an example of symmetry
Here is a paragraph about the transformations I used to create my logo.
How to explain the transformationI started with a basic shape, a square. I then reflected the square across the horizontal axis, creating a mirror image. I then translated the reflected square up and to the right, creating a new shape. I then rotated the new shape 45 degrees, creating a third shape. The third shape is symmetric, meaning that it can be divided into two identical halves.
I then used these three shapes to create my logo. I used the first shape as the background, the second shape as the foreground, and the third shape as the logotype. I chose these shapes because they represent the three stages of transformation: the beginning, the middle, and the end.
The reflection represents the beginning of the transformation, when something is changed from its original state. The translation represents the middle of the transformation, when something is in the process of being changed. The rotation represents the end of the transformation, when something has been completely changed.
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for this problem, use the data from part (a). if we consider only the people who named a rainbow color (red, orange, yellow, green, blue, or purple), what percentage of those people preferred red?
To answer this question, we need to look at the data from part (a) and only consider the people who named a rainbow color. Out of those people, we can see that 35% preferred red.
To calculate the percentage, we can divide the number of people who preferred red (14) by the total number of people who named a rainbow color (40), and then multiply by 100.
So, (14/40) x 100 = 35%.
Therefore, 35% of the people who named a rainbow color preferred red. It's important to note that this percentage only applies to the subset of people who named a rainbow color, not the entire group of people surveyed.
Overall, it's always important to clarify the specific subset of data you're analyzing when discussing percentages or statistics.
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You roll a standard die (with the numbers {1,2,3,4,5,6} on its faces). Let A be the event that you roll an odd number. What is the probability of the event A?
Give your answer as a fraction.
[tex]|\Omega|=6\\|A|=3\\\\P(A)=\dfrac{3}{6}=\dfrac{1}{2}[/tex]
Answer:
Step-by-step explanation: A represents 1, 3, 5, or a half of the numbers.
3/6 reduces to 1/2 final answer.
how many triangles can be formed by connecting three points out of nine on the circumference of a circle
The number of triangles can be formed 84
Triangle:A triangle is a polygon with three sides which consists of three vertices. A triangle can be formed from three given points only if the three points do not lie on a straight line and also form 180 degree angle.
Since the 3 points are distinct and lie on a circle, then it is not possible to have three points such that all three lie on a line.
Hence, we can use these 3 points to form triangles by selecting 3 points. The number of such ways is the number of different triangles we can form. Hence,
Triangles = [tex]C^9_3[/tex]
Triangles = [tex]\frac{9!}{3!(6!)}[/tex]
Triangles = 84
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the diagonals of a rhombus measure 18 feet and 12 feet. what is the perimeter of the rhombus? express your answer in simplest radical form
The perimeter of the rhombus is 4√117 feet.
What is rhombus?
A rhombus is a special type of quadrilateral (a polygon with four sides) where all four sides are equal in length. It is also known as a diamond shape because of its symmetrical appearance.
In a rhombus, the diagonals are perpendicular bisectors of each other. This means that they divide the rhombus into four congruent right triangles.
Let's denote the length of one diagonal as d1 and the length of the other diagonal as d2. In this case, d1 = 18 feet and d2 = 12 feet.
Each right triangle formed by the diagonals has legs equal to half the lengths of the diagonals. Therefore, the lengths of the legs are d1/2 = 18/2 = 9 feet and d2/2 = 12/2 = 6 feet.
Using the Pythagorean theorem, we can find the length of the hypotenuse of each right triangle, which is also a side of the rhombus.
[tex]h^2 = (leg1)^2 + (leg2)^2\\\\h^2 = 9^2 + 6^2\\\\h^2 = 81 + 36\\\\h^2 = 117[/tex]
Taking the square root of both sides:
h = √117
Since the rhombus has four congruent sides, the perimeter is given by:
Perimeter = 4 * h = 4 * √117
Thus, the perimeter of the rhombus is 4√117 feet.
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Tim brought $40. 50 to the state fair. He bought a burger, a souvenir, and a pass. The burger was
1
4
as much as the souvenir, and the souvenir cost
2
3
the cost of the pass. Tim had $2. 00 left over after buying these items. What was the cost of each item?
The cost of the burger was approximately $2.19, the cost of the souvenir was approximately $8.
let's first assign variables to the cost of each item. let b be the cost of the burger, s be the cost of the souvenir, and p be the cost of the pass.
from the problem, we know that:
b + s + p = 40.50 (total cost of all three items)b = (1/4)s (the burger cost one-fourth the price of the souvenir)
s = (2/3)p (the souvenir cost two-thirds the price of the pass)b + s + p + 2.00 = 40.50 (tim had $2.00 left over)
we can use substitution to solve for the unknown variables.
substituting b = (1/4)s and s = (2/3)p into the first equation, we get:
(1/4)s + s + (2/3)p = 40.50
multiplying both sides by 12 to eliminate the fractions, we get:
3s + 12s + 8p = 486
simplifying, we get:
15s + 8p = 486
substituting s = (3/2)p into the above equation, we get:
15(3/2)p + 8p = 486
simplifying further, we get:
37p = 486
solving for p, we get:
p = 486/37
p ≈ 13.14
now that we know the cost of the pass, we can use the second equation to find the cost of the souvenir:
s = (2/3)p = (2/3)(13.14) ≈ 8.76
finally, we can use the first equation to find the cost of the burger:
b = (1/4)s = (1/4)(8.76) ≈ 2.19 76, and the cost of the pass was approximately $13.14.
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Pls answer 40pts Drag the tiles to the correct boxes to complete the pairs.
Match each quadratic function to its graph.
f(x) = -2(x + 3)2 − 1
f(x) = -2(x + 3)2 + 1
f(x) = 2(x + 3)2 + 1
f(x) = 2(x − 3)2 + 1
Answer:
Step-by-step explanation:
f(x) = -2(x+3)^2 -1 would be the fourth graph because its translated 3 to the left, negative, and 1 down
f(x)=2(x+3)^2+1 would be the first graph since it's translated 3 to the left, positive, and shifted 1 up
f(x)=-2(x+3)^2+1 would be the second graph since it's translated 3 to the right, negative, and shifted 1 up
f(x)=2(x-3)^2+1 would be the third graph since it's translated 3 to the right, positive, and shifted 1 up
