Anne is taking courses in both mathematics and English. She estimates her probability of passing mathematics at 0. 42 and passing English at 0. 47 , and she estimates her probability of passing at least one of the courses at 0. 7. What is the probability that Anne could pass both courses?

For approximately 64% of the shows, the theater was half full or less than half full.

Since the seat capacity of the theater is 300, half full would be 150 seats or less. Looking at the histogram, we can see that there are 3 bars representing showings with 150 or less viewers.

The first bar represents showings with 0-50 viewers. From the histogram, it looks like there were about 5 showings with this number of viewers.

The second bar represents showings with 50-100 viewers. From the histogram, it looks like there were about 8 showings with this number of viewers.

The third bar represents showings with 100-150 viewers. From the histogram, it looks like there were about 3 showings with this number of viewers.

So the total number of showings with 150 or less viewers is 5+8+3 = 16.

Since the histogram shows a total of 25 showings, the fraction of shows that was half full or less than half is:

16/25 = 0.64 or 64%

Therefore, for approximately 64% of the shows, the theater was half full or less than half full.

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Therefore, the solutions of tan x = -1/2 in the interval (0, 2) are:

x ≈ 2.034, 5.176

We can simplify the given equation as follows:

4 tan 2x - 4 cot x = 0

4(tan 2x - cot x) = 0

4[(2tan x)/(1 - tan^2 x) - (1)/(tan x)] = 0

Multiplying both sides by (1 - tan^2 x) * (tan x), we get:

8tan^3 x - 4tan^2 x - 8tan x + 4 = 0

Dividing both sides by 4 and rearranging, we get:

2tan^3 x - tan^2 x - 2tan x + 1 = 0

Factorizing, we get:

(tan x - 1)(2tan^2 x - tan x - 1) = 0

Using the quadratic formula to solve for the roots of 2tan^2 x - tan x - 1 = 0, we get:

tan x = [1 ± sqrt(1 + 8)] / 4 = [1 ± sqrt(9)] / 4 = 1, -1/2

Therefore, the solutions of the given equation in the interval (0, 2) are the values of x such that tan x = 1 or tan x = -1/2.

We know that tan (π/4) = 1 and tan (-π/4) = -1, so the solutions of tan x = 1 in the interval (0, 2) are:

x = π/4, 5π/4

We can find the solutions of tan x = -1/2 in the interval (0, 2) by finding the reference angle and using the signs of sine and cosine in the corresponding quadrants. We have:

tan x = -1/2

Let θ be the reference angle such that tan θ = 1/2. We know that θ is in the second or fourth quadrant.

In the second quadrant, sine is positive and cosine is negative, so we have:

sin θ = sqrt(1/(1 + tan^2 θ)) = sqrt(1/5)

cos θ = -tan θ = -1/2

Therefore, we get:

x = π - θ = π + arctan(1/2) ≈ 2.034

In the fourth quadrant, both sine and cosine are negative, so we have:

sin θ = -sqrt(1/(1 + tan^2 θ)) = -sqrt(1/5)

cos θ = -tan θ = -1/2

Therefore, we get:

x = 2π - θ = 2π + arctan(1/2) ≈ 5.176

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The ball will rebound to maximum height of 25 centimetres or 0.25 meters after 7 bounces.

Firstly perform the unit conversion. As known, 1 meter is 100 cm. So, 25 centimetres is 0.25 meters.

Now, the formula to be used to find the number of bounces is -

New height × [tex] {2}^{n} [/tex] = old height, where n refers to number of bounces.

Keeping the values in formula

0.25 × [tex] {2}^{n} [/tex] = 32

Rearranging the equation

[tex] {2}^{n} [/tex] = 32/0.25

Divide the values

[tex] {2}^{n} [/tex] = 128

Converting the result into exponent form

[tex] {2}^{n} [/tex] = 2⁷

Thus, n will be 7 bounces.

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The scale factor of figure KLMN to figure PQRS is 6 cm / 3 cm = 2.

To find the scale factor of figure KLMN to figure PQRS, given KN = 3 cm, MN = 7 cm, RS = 14 cm, and PS = 6 cm, follow these steps:

1. First, find the length of a side in figure KLMN. We can use KN since it's given: KN = 3 cm.
2. Next, find the corresponding side length in figure PQRS. Since KN corresponds to PS, we have: PS = 6 cm.
3. Now, find the scale factor by dividing the length of the side in figure PQRS by the length of the corresponding side in figure KLMN: scale factor = PS/KN = 6 cm / 3 cm.

The scale factor of figure KLMN to figure PQRS is 6 cm / 3 cm = 2.

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The circle has horizontal tangent lines at the points (1, 2 + √(14)) and (1, 2 - √(14)) and the circle has vertical tangent lines at the points (1 + √(8), -2) and (1 - √(8), -2).

To find the points where the circle defined by x^2 + y^2 - 2x - 4y = 11 has horizontal and vertical tangent lines, we need to use implicit differentiation to find the derivatives of x and y with respect to each other.

Taking the derivative of both sides of the equation with respect to x, we get:

2x + 2y(dy/dx) - 2 - 4(dy/dx) = 0

Simplifying, we get:

(dy/dx) = (x-1) / (y+2)

To find the points where the circle has horizontal tangent lines, we need to find where the derivative dy/dx is equal to zero. This occurs when x-1 = 0, or x = 1. Substituting this value of x back into the original equation, we get:

1 + y^2 - 4y = 11

Simplifying, we get:

y^2 - 4y - 10 = 0

Using the quadratic formula, we get:

y = 2 ± √(14)

To find the points where the circle has vertical tangent lines, we need to find where the derivative dy/dx is undefined, which occurs when y+2 = 0, or y = -2. Substituting this value of y back into the original equation, we get:

x^2 - 2x + 4 = 11

Simplifying, we get:

x^2 - 2x - 7 = 0

Using the quadratic formula, we get:

x = 1 ± √(8)

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The volume of the larger box is 200 cubic inches, and the volume of the smaller box is 150 cubic inches.

To find the volume of each box, we use the formula for the volume of a rectangular prism, which is V = lwh, where l is the length, w is the width, and h is the height.

For the larger box, we have l = 8 inches, w = 2.5 inches, and h = 10 inches, so

Volume of large box = = 8 x 2.5 x 10 = 200 cubic inches.

For the smaller box, we have l = 0.75 x 8 = 6 inches, w = 2.5 inches, and h = 10 inches, so

Volume of small box= 6 x 2.5 x 10 = 150 cubic inches.

The difference in the volumes of the two boxes is

Volume of large box - Volume of small box = 200 - 150 = 50 cubic inches.

The units for the volumes are cubic inches, since we are dealing with three-dimensional space.

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The point estimate for μ is 15.3 years, based on the sample of 60 Grade 9 students.

Based on the statistical techniques given information, the point estimate for the population mean age of all Grade 9 students is 15.3 years. This means that if we assume that the sample is representative of the entire population of Grade 9 students, then we estimate that the average age of all Grade 9 students is 15.3 years.

To estimate the precision of this point estimate, we can calculate confidence intervals. For a 95% confidence interval, we can use the formula:

CI = X ± (Zα/2) * (σ/√n)

where X is the point estimate, Zα/2 is the critical value of the standard normal distribution for a 95% confidence level (1.96), σ is the population standard deviation (which we assume to be known as 4), and n is the sample size (which is 60).

Substituting the values, we get the 95% confidence interval as:

CI = 15.3 ± (1.96) * (4/√60) = (14.33, 16.27)

This means that we can be 95% confident that the true population mean age of Grade 9 students lies between 14.33 and 16.27 years.

For a 99% confidence interval, we can use the same formula with a different value of Zα/2 (2.58 for a 99% confidence level). Substituting the values, we get the 99% confidence interval as:

CI = 15.3 ± (2.58) * (4/√60) = (13.94, 16.66)

This means that we can be 99% confident that the true population mean age of Grade 9 students lies between 13.94 and 16.66 years.

Based on the confidence intervals, we can conclude that the sample provides evidence that the true mean age of all Grade 9 students is likely to be between 14.33 and 16.27 years with a 95% confidence level, and between 13.94 and 16.66 years with a 99% confidence level. However, we cannot be completely certain that the true population mean falls within these intervals as there is always some level of uncertainty associated with sample-based estimates.

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The area of ΔOPQ is approximately 18.9 square inches, rounding off to nearest 10th.

To find the area of ΔOPQ, we can use the formula:

Area = (1/2) * base * height

We know that p = 9.5 inches, q = 7.6 inches, and ∠O = 31°.

Now, using  trigonometry the height h of the triangle can be found using the sin function.

              sin(θ) = perpendicular/hypotenuse

perpendicular = hypotenuse* sin(θ)

                        = 7.6 * sin(31°)

                        ≈ 3.98 inches

Now, we can use the formula for the area:

Area = (1/2) * base * height

Putting in the values, we get:

Area = (1/2) * 9.5 * 3.98

Area ≈ 18.93 square inches

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For water [OH⁻] equals 8.70 x 10⁻²³ M when [H⁺] = 1.15 x 10⁸ M .

Assuming that the ion product constant for water is being questioned, Kw:

Kw = [H⁺] [OH⁻]

where Kw = ion product constant for water

kW at 25°C = 1.0 x 10⁻¹⁴

Since Kw is constant at a given temperature, if we know the concentration of [H⁺] or [OH⁻], we can find it with Kw the concentration of the new ion

given that [H⁺] = 1.15 x 10⁸ M, [OH⁻] can be determined as follows.

Kw = [H⁺] [ OH⁻]

plugging in the values

1.0 x 10⁻¹⁴ M² = (1.15 x 10⁸ M) * ([OH⁻])

[OH⁻] = (1.0 x 10⁻¹⁴ M² ) / (1.15 x 10⁸ M )

[OH⁻] = 8.70 x 10⁻²³ M

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Answer:

Isolate the variable by dividing each side by factors that don't contain the variable.

Exact Form:

x = −1/2

Decimal Form:

x = −0.5

Step-by-step explanation:

Answer:

x = 24

Step-by-step explanation:

using the rule of logarithms

[tex]log_{b}[/tex] x = n ⇒ x = [tex]b^{n}[/tex]

note that log x represents [tex]log_{10}[/tex] x

given

log(4x + 4) = 2 , then

4x + 4 = 10² = 100 ( subtract 4 from both sides )

4x = 96 ( divide both sides by 4 )

x = 24

Laptop- 44, Tablet- 94, Desktop- 62, Percentage of tablets- 47%.

To calculate the percentage of devices that are tablets, we need to find the proportion of tablets among all devices. In this case, there are 94 tablets out of a total of 200 devices (44 laptops + 94 tablets + 62 desktops).

To find the proportion, we divide the number of tablets by the total number of devices: 94/200 = 0.47.

To convert this proportion to a percentage, we multiply by 100: 0.47 * 100 = 47%.

47% of the devices owned by the local company are tablets.

This means that nearly half of the devices fall into the tablet category, making it a significant portion of their device inventory.

It's important to note that this calculation assumes the given numbers accurately represent the actual device distribution and that there are no missing or unaccounted devices.

By determining the percentage of tablets, we gain insight into the composition of the device inventory, which can be useful for decision-making and resource allocation within the local company.

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Answer is 7/3 cups.

To determine the amount of raisins needed for the snack mix, subtract 3 5/12 cups from 5 3/4 cups of cereal.

First, convert the mixed numbers to improper fractions:
5 3/4 = (5 × 4 + 3)/4 = 23/4
3 5/12 = (3 × 12 + 5)/12 = 41/12

Next, subtract the two fractions:
23/4 - 41/12

To subtract, find a common denominator. The least common multiple of 4 and 12 is 12. Convert both fractions to equivalent fractions with a denominator of 12:
(23/4) × (3/3) = 69/12
(41/12) × (1/1) = 41/12

Now, subtract the fractions:
69/12 - 41/12 = 28/12

Finally, simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (4):
28/12 = (28 ÷ 4)/(12 ÷ 4) = 7/3

So, you need 7/3 cups of raisins for the snack mix.

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Answer:

Step-by-step explanation:

where are the streets

The value of median m that makes Q₃ equal to 128 is approximately 18.75.

The median is the value that divides the higher half of a population, a probability distribution, or a sample of data from the lower half. It can be conceptualised as a data set's "middle" value to put it simply.

To find the value of m, we need to first calculate the median and third quartile of the data.

To calculate the median, we need to find the value that splits the data into two halves. Since the data is already sorted into intervals, we can find the cumulative frequency for each interval and use it to determine the median interval. The median interval is the interval that contains the median. We can then use the formula for the median of grouped data to calculate the median value.

Cumulative frequency for each interval:

- Interval 0-30: 2

- Interval 30-60: 2+8=10

- Interval 60-90: 10+22=32

- Interval 90-120: 32+24=56

- Interval 120-150: 56+m

- Interval 150-180: 56+m+9=65+m

Since there are 6 intervals, the median interval is the 3rd interval, which is 60-90. The lower limit of this interval is 60, and the cumulative frequency up to this interval is 32. The frequency of this interval is 22. Using the formula for the median of grouped data:

Median = L + ((n/2 - CF) / f) * w

where L is the lower limit of the median interval, CF is the cumulative frequency up to the median interval, n is the total sample size, f is the frequency of the median interval, and w is the width of the interval.

Plugging in the values, we get:

Median = 60 + ((50 - 32) / 22) * 30

Median = 60 + (18 / 22) * 30

Median = 60 + 15.45

Median ≈ 75.45

Now, to find the third quartile (Q₃), we need to find the value that splits the upper 50% of the data. Since Q₃ is the 75th percentile, the cumulative frequency up to Q₃ is 0.75 times the total sample size:

Q₃ = L + ((0.75 * n - CF) / f) * w

We know that Q₃ is 128, and we can plug in the values for L, n, CF, f, and w that correspond to the interval that contains Q₃:

128 = 120 + ((0.75 * 85 - 56 - m) / (24)) * 30

Simplifying and solving for m, we get:

m = 120 + ((0.75 * 85 - 56) / (24)) * 30 - 128

m ≈ 18.75

Therefore, the value of m that makes Q₃ equal to 128 is approximately 18.75.

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Answer:

(5.5, -1.5)

Step-by-step explanation:

(x, y)midpoint = (x1 + x2)/2 , (y1 + y2)/2

= (6 + 5)/2, (3 - 6)/2

= (11/2, -3/2)

= (5.5, -1.5)

The inequality which can be used to determine amount the club needs to raise during remaining months is 400 + 4n ≥ 750.

The goal of the school chess club is to raise at least $750 in total so that they can attend a state competition. They have already raised $400, but they still need to raise more money. Let's call the amount they need to raise each month "n".

Since the club has 4 months remaining until the competition, they will need to raise a total of "4n" dollars during that time period.

To determine the minimum amount they need to raise each month, the inequality can be written as : 400 + 4n ≥ 750,

4n ≥ 350 ; n ≥ 87.5.

This means that the chess-club needs to raise at least $87.50 each month in order to reach their goal of $750 in 4 months.

Therefore, the required inequality is 400 + 4n ≥ 750.

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The given question is incomplete, the complete question is

A school chess club needs to raise at least $750 to attend a state competition. The club has already raised $400 and there are 4 months remaining until the competition. Write an inequality which can be used to determine the dollar amount the club will need to raise during the remaining months?

The zero of the function is 14. In the context of the situation, this means it will take 14 hours for Libby to completely drain the pool for cleaning.

To find the zero of the function, we need to determine the value of x when y equals 0.

0 = 10,080 - 720x

To solve for x, we will isolate the variable by following these steps:

1. Add 720x to both sides:
720x = 10,080

2. Divide both sides by 720:
x = 14

The zero of the function is 14, which means that after 14 hours, there will be no water remaining in the pool. In the context of the situation, this means it will take 14 hours for Libby to completely drain the pool for cleaning.

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Answer:

--------------------

Let the original number be n.

We are given that 30% of n is 14.99.

Set up an equation to reflect this relation:

Solve it for n:

The demand equation can be used to estimate the demand for air travel at different price levels, and can help airlines make pricing decisions based on the expected demand.

The demand equation is given as:

q = 55.1 - 0.023p

where q is the quantity demanded and p is the price of the airfare.

This equation shows an inverse relationship between price and quantity demanded. As the price of the airfare increases, the quantity demanded decreases, and vice versa.

For example, if the airfare price is $100, we can calculate the quantity demanded as:

q = 55.1 - 0.023(100) = 52.8

This means that at a price of $100, the quantity demanded is approximately 52.8 units.

Similarly, if the airfare price is $200, we can calculate the quantity demanded as:

q = 55.1 - 0.023(200) = 50.4

This means that at a price of $200, the quantity demanded is approximately 50.4 units.

So, demand equation can be used to estimate the demand for air travel at different price levels, and can help airlines make pricing decisions based on the expected demand.

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As per the growth function, the value of the quantity after 97 years would be $67,458.85.

In your problem, you have a quantity with an initial value of 8200 that grows continuously at a rate of 0.55% per decade. To find the value of the quantity after 97 years, we can use the following growth function:

A(t) = A₀[tex]e^{kt}[/tex]

In this formula, A(t) represents the value of the quantity after time t, A₀ represents the initial value of the quantity (in this case, 8200), e represents Euler's number (a mathematical constant equal to approximately 2.718), k represents the growth rate (in this case, 0.0055 per decade), and t represents the time elapsed (in this case, 97 years).

To solve for the value of the quantity after 97 years, we simply plug in the values we know and solve for A(t):

A(t) = 8200[tex]e^{(0.0055/10\times97)}[/tex]

= 8200[tex]e^{0.5285}[/tex]

≈ 67,458.85

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The size of the square that needs to be cut out of each corner to create a box of maximum volume is 5/3 inches.

To determine the function f given that f'(x)=6x² – 5 sin x+eˣ and f(0) = 20, we need to integrate f'(x) with respect to x to obtain f(x), and then use the initial condition f(0) = 20 to find the value of the constant of integration.

Integrating f'(x) with respect to x, we have:

f(x) = 2x³ + 5 cos x + eˣ + C

where C is the constant of integration.

Using the initial condition f(0) = 20, we have:

f(0) = 2(0)³ + 5 cos 0 + e⁰ + C = 6 + C = 20

Therefore, the constant of integration is C = 14, and the function f(x) is:

f(x) = 2x³ + 5 cos x + eˣ + 14

To determine the size of the square that needs to be cut out of each corner of a standard piece of notebook paper to create a box of maximum volume, we can start by drawing a diagram of the box and labeling the sides as follows:

| |

| |

| | h

| |

|__________|

L

Let x be the length of each side of the square that is cut out of each corner. Then, the length and width of the base of the box will be L - 2x and 11 - 2x, respectively, and the height of the box will be x. Therefore, the volume V of the box can be expressed as:

V(x) = x(L - 2x)(11 - 2x)

Expanding and simplifying, we get:

V(x) = -4x³ + 46x² - 110x

To find the size of the square that maximizes the volume of the box, we need to find the value of x that maximizes V(x). This can be done by finding the critical points of V(x) and determining whether they correspond to a maximum or minimum.

Taking the derivative of V(x) with respect to x, we get:

V'(x) = -12x² + 92x - 110

Setting V'(x) = 0 and solving for x, we get:

x = 5/3 or x = 11/6

To determine whether these values correspond to a maximum or minimum, we can use the second derivative test. Taking the second derivative of V(x) with respect to x, we get:

V''(x) = -24x + 92

Evaluating V''(5/3) and V''(11/6), we find that:

V''(5/3) = -4 < 0, so x = 5/3 corresponds to a maximum.

V''(11/6) = 20 > 0, so x = 11/6 corresponds to a minimum.

Therefore, the size of the square that needs to be cut out of each corner to create a box of maximum volume is 5/3 inches.

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The correct expressions are:

A) sin θ = y

B) tan θ = y/x

C) cos θ = x

On a unit circle, the terminal side of Angle 0 intersects the circle at points (x,y). We can use trigonometric ratios to relate the coordinates (x,y) to the angle θ.

A) sin θ = the ratio of y to 1, or simply y.

B) tan θ = the ratio of y to x.

C) cos θ = the ratio of x to 1, or simply x.

Therefore, the correct expressions are:

A) sin θ = y

B) tan θ = y/x

C) cos θ = x

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The total number of students in Mrs. Cunningham's class is 30.

From the question we know that the ratio of boys to girls in Mrs. Cunningham's class is 2 to 3 so we can write

no.of boys: no.of girls = 2:3

The total number of girls in the class is given as 18 so with this we can find out the number of boys in the class that is :

no.of boys= (2/3)*no.of girls in class

now after substituting the values in the equation, we get

no. of boys = (2/3) * 18

no.of boys = 12.

So, now we know the number of boys in the class that is 12 and the number of girls in the class is 18.

We can calculate the total number of students in the class which is equal to

= no.of boys + no.of girls.

= 12+18

=30

Therefore, the total number of students in Mrs. Cunningham's class is 30.

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The given equation for the value of the rocking chair is V = 115(1.6)^t, where t is the number of years since 1975. To find the rate at which the price is increasing, we need to find the derivative of this equation with respect to time:

dV/dt = 115(1.6)^t * ln(1.6)

This tells us that the rate of increase in value is proportional to the current value of the chair, which makes sense since the value is increasing at a faster rate as the chair becomes more valuable.

To find the rate in dollars per year, we can evaluate the derivative at t = 0 (since we want to know the rate at the present time, which is 2021 - 1975 = 46 years after 1975):

dV/dt = 115(1.6)^0 * ln(1.6) = 30.03

Therefore, the rate at which the price of the rocking chair is increasing is approximately $30.03 per year.
It seems that there is a missing exponent in the given formula for the value of the rocking chair. The correct formula should include an exponent 't' as in V = 115(1.6)^t, where V is the value in dollars and t is the number of years since 1975.

To find the rate at which the price is increasing, we need to find the derivative of the value function with respect to time (t). The derivative of V = 115(1.6)^t is dV/dt = 115 * ln(1.6) * (1.6)^t.

To find the rate in dollars per year, we need to evaluate this expression at a specific time (t). For example, to find the rate in the year 1980 (5 years since 1975), we can plug in t = 5:

Rate = 115 * ln(1.6) * (1.6)⁵ ≈ $419.20 per year

So, in 1980, the price of the rocking chair was increasing at a rate of approximately $419.20 per year.

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Answer:

0.91

30.4

850

Step-by-step explanation:

x² + 10x + 25 is the expression of (x+5)^2 in trinomial in standard form

A trinomial is a polynomial that consists of three terms. It is a type of algebraic expression that contains three algebraic terms separated by either plus or minus signs.

For example, the expression 2x^2 + 5x - 3 is a trinomial because it has three terms: 2x^2, 5x, and -3. Similarly, the expression 4a^3 - 7a^2 + 2a is also a trinomial because it has three terms: 4a^3, -7a^2, and 2a.

We have to expand this

x² + 2(x)(5) + 5²

= x² + 10x + 25

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Answer 1: Together, Streetcar and Cable Car are the preferred transportation for 168 residents.

The circle graph shows the percentage of residents who prefer each transportation method, and the total sample size is 400.

For streetcar, (15/100) x 400 = 60 residents prefer it, and for cable car, (27/100) x 400 = 108 residents prefer it.

Together, Streetcar and Cable Car are the preferred transportation for 60 + 108 = 168 residents.

Answer 2: The median is the best measure of center, and it equals 19.

The box plot shows the distribution of the number of tickets sold for a school dance.

The median is the middle value of the data when arranged in order, and it is represented by the line in the box. In this case, the median is 19. The mean, on the other hand, can be influenced by extreme values, and we cannot determine it from the box plot alone.

Answer 3: Median, because Sunny Town is skewed.

When comparing the data, we need to consider the measure of center that is less affected by extreme values, and that is the median.

The median is the middle value of the data when arranged in order. The histogram for Sunny Town is skewed to the right, which means that there are some very high values that are affecting the mean.

Therefore, the median is the better measure of center to determine which location typically has the cooler temperature.

Answer 4: The IQR of 13 is the most accurate to use, since the data is skewed.

The histogram shows the frequency of donations received by a charity, and the data is skewed to the right.

The IQR (Interquartile Range) is the difference between the third quartile (Q3) and the first quartile (Q1), which represents the middle 50% of the data.

The IQR is less sensitive to extreme values and is a better measure of variability for skewed data. In this case, the IQR is 49 - 42 = 7, which is the most accurate measure of variability to use.

Answer 5: There were about 15 principals in attendance.

In the exhibit room, out of 80 people, 15 are principals.

We can assume that the proportion of principals in the exhibit room is the same as the proportion of principals in the conference.

Therefore, the estimated number of principals in the conference is (15/80) x 900 = 168.75, which is approximately 169.

Answer 6: Histogram

The teacher wants to represent the subject preferences of 100 students. A histogram would be the best graphical representation to use because it shows the frequency distribution of a continuous variable, which in this case could be the number of students who prefer each subject.

A stem-and-leaf plot is used for small datasets, and a box plot is used to display the distribution of a continuous variable across categories. A circle graph is more appropriate for displaying categorical data, such as the percentage of students who prefer each subject.

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To find u × v, we use the cross product formula:

u × v = | i    j    k |
           | 2    0    6 |
           | 4    7   -5 |

Expanding the determinant, we get:

u × v = (0*-5 - 6*7) i - (2*-5 - 6*4) j + (2*7 - 0*4) k
u × v = -42i - 22j + 14k

To find v × u, we use the same formula but switch the order of u and v:

v × u = | i    j    k |
           | 4    7   -5 |
           | 2    0    6 |

Expanding the determinant, we get:

v × u = (7*6 - (-5)*0) i - (4*6 - (-5)*2) j + (4*0 - 7*2) k
v × u = 42i + 18j - 14k

Finally, to find v × v, we again use the cross product formula with v as both inputs:

v × v = | i    j    k |
           | 4    7   -5 |
           | 4    7   -5 |

Expanding the determinant, we get:

v × v = (7*(-5) - (-5)*7) i - (4*(-5) - (-5)*4) j + (4*7 - 7*4) k
v × v = 0i - 0j + 0k
v × v = 0

So the cross product of v with itself is the zero vector.
To find u × v, v × u, and v × v, we'll use the cross product formula:

u × v = (u_yv_z - u_zv_y)i + (u_zv_x - u_xv_z)j + (u_xv_y - u_yv_x)k

Given u = 2i + 6k and v = 4i + 7j - 5k, we have:

u_x = 2, u_y = 0, u_z = 6
v_x = 4, v_y = 7, v_z = -5

Now, calculate u × v:
(0 * (-5) - 6 * 7)i + (6 * 4 - 2 * (-5))j + (2 * 7 - 0 * 4)k
= (-42)i + (34)j + (14)k

u × v = -42i + 34j + 14k

Next, calculate v × u:
(7 * 6 - (-5) * 0)i + ((-5) * 2 - 4 * 6)j + (4 * 0 - 7 * 2)k
= (42)i + (-34)j + (-14)k

v × u = 42i - 34j - 14k

Finally, calculate v × v:
(7 * (-5) - (-5) * 7)i + ((-5) * 4 - 4 * (-5))j + (4 * 7 - 7 * 4)k
= (0)i + (0)j + (0)k

v × v = 0i + 0j + 0k

In summary:
u × v = -42i + 34j + 14k
v × u = 42i - 34j - 14k
v × v = 0i + 0j + 0k

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