See the photo below

Answer:

Step-by-step explanation:

The distance formula is

[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]

x1 is a,

x2 is 0,

y1 is 0 and

y2 is b. Fitting those into the formula where they belong:

[tex]d=\sqrt{(0-a)^2+(b-0)^2}[/tex] and

[tex]d=\sqrt{(-a)^2+(b)^2}[/tex]

Since a negative squared is a positive, then

[tex]d=\sqrt{a^2+b^2}[/tex]

which is the second choice down.

The exponential function of the population is P(x) = 15000 * 1.046^x

From the question, we have the following parameters that can be used in our computation:

Initial, a = 15000

Rate, r = 4.6%

The equation of the function is represented as

P(x) = a * (1 + r)^x

Substitute the known values in the above equation, so, we have the following representation

P(x) = 15000 * (1 + 4.6%)^x

Evaluate

P(x) = 15000 * 1.046^x

Hence, the function is P(x) = 15000 * 1.046^x

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Answer:

1.2974634e+14

Step-by-step explanation:

ye

We need to collect 46 newspapers to be certain that at least two newspapers have the same number of pages.

According to the Pigeonhole Principle, if we have n+1 pigeons and n holes, then there must be at least one hole with two or more pigeons. Similarly, if we have n+1 newspapers with n possible page counts, then there must be at least one page count that appears in two or more newspapers.

In this case, we have a range of 38 possible page counts (46 - 9 + 1), so we need at least 39 newspapers to guarantee that each possible page count appears in at least one newspaper.

However, to guarantee that at least two newspapers have the same number of pages, we need one more newspaper than the number of possible page counts, so we need a total of 46 newspapers. Therefore, if we collect 46 newspapers, we can be certain that at least two of them have the same number of pages.

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Answer:

18 5-dollar bills

10 20-dollar bills

By answering the presented question, we may conclude that As a result, adding the isosceles triangle categorization is superfluous and redundant in this circumstance.

A triangle is a polygon since it has three sides and three vertices. It is a basic geometric shape. Triangle ABC refers to a triangle with the vertices A, B, and C. In Euclidean geometry, a single plane and triangle are obtained when the three points are not collinear. If a triangle has three sides and three corners, it is a polygon. The triangle's corners are the spots where the three sides meet. The sum of three triangle angles equals 180 degrees.

An isosceles triangle has two sides of equal length, but a right triangle has one angle that measures 90 degrees. As a result, an isosceles right triangle is one with two equal-length sides and one 90-degree angle.

The Pythagorean theorem states that in any right triangle, the two legs (the sides next to the right angle) are always of equal length. Hence, if we know that a triangle has one 90-degree angle and two equal-length sides, we already know that it is a right triangle and that the two legs are equal. As a result, adding the isosceles triangle categorization is superfluous and redundant in this circumstance.

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Answer:

Step-by-step explanation:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,144,233,377,610,987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, ... Can you figure out the next few numbers?

Therefore, the correct answer is: The slope of g(x) is less than the slope of f(x).

The origin is the location where the two axes meet. On both the x- and y-axes, the origin is at 0. The coordinate plane is divided into four portions by the intersection of the x- and y-axes. The term "quadrant" refers to these four divisions.

We can use the slope formula to get the slopes of the lines f(x) and g(x):

slope of f(x) = (change in y)/(change in x) = (1 - (-2))/(1 - 0) = 3/1 = 3

slope of g(x) = (change in y)/(change in x) = (2 - 0)/(0 - (-4)) = 2/4 = 1/2

The slope of g(x) is 1/2, which is less than the slope of f(x), which is 3.

Therefore, the correct answer is: The slope of g(x) is less than the slope of f(x).

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Expressions for the square footage of the tile Dining area, Considering only the tile floor, and not the cabinets shown in black will be (16 x 13) - (12 x 2) and (4 x 13) + (12 x 11)

The area of a rectangle is a measure of the amount of space it occupies in two-dimensional (2D) space. It is calculated by multiplying the length of the rectangle by its width. Mathmatically,

                                     [tex]Area=length*width[/tex]

Now, Solving given problem,

The total area of the grid will be:Length of grid = 16 ft, Width of grid = 13 ft

Total area of grid = Length x Width = 16 ft x 13 ft = 208 sq ft

The area of the rectangle in the top right :Length of rectangle = 12 ft, Width of rectangle = 2 ft

Area of rectangle = Length x Width = 12 ft x 2 ft = 24 sq ft

To remove the top right rectangle from the tile dining area, subtract its area from the total area of the grid:

(Total area of grid) - (Area of rectangle in top right) = (208 sq ft -24 sq ft )= 184 sq ft

So, the expression for this calculation will be: (16 x 13) - (12 x 2) = 184 sq ft

The area of the vertical rectangles on the far left and right sides is calculated as follows:Width of vertical rectangles = 4 ft, Length of grid = 13 ft

Area of vertical rectangles = Width of vertical rectangles x Length of grid = 4 ft x 13 ft = 52 sq ft

The area of the rectangle in the top right:Length of rectangle = 12 ft, Width of rectangle = 11 ft

Area of rectangle = Length x Width = 12 ft x 11 ft = 132 sq ft

Add the areas of the vertical rectangles and the rectangle in the top right to get the total area of the tile dining area:

Area of vertical rectangles + Area of rectangle in top right = 52 sq ft + 132 sq ft = 184 sq ft

So, the expression for this calculation is: (4 x 13) + (12 x 11) = 184 sq ft

Hence, both of these expressions- (16 x 13) - (12 x 2) and (4 x 13) + (12 x 11) gives the square footage of the tile dining area based on the given information.

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Answer: The most appropriate design for this experiment is the third option:

- Number each item from 0000 to 1199.

- Reading from left to right on a random number table, identify 800 unique four-digit numbers from 0000 to 1199. Assign the items with labels in the first 400 numbers to the process 1 group. Assign the items with labels in the second 400 numbers to the process 2 group. Assign the remaining items to the process 3 group.

This design ensures that the groups are equally sized and selected randomly without any biases. The use of a random number table to assign the groups helps to avoid any systematic patterns or preferences that might arise from numbering or labeling the items directly.

Step-by-step explanation:

The formula g(x) = 2 * 0{-10≤x+3≤10} represents a function that has been horizontally shifted left by 3 units, reflected about the y-axis, and vertically scaled by a factor of 2.

A function is a mathematical rule that assigns each input from a set (domain) a unique output from another set (range), typically written as y = f(x).

The notation "0{-10≤x≤10}" typically represents the domain of a function or an inequality. It means that the function is defined only for the values of x that are between -10 and 10 (including -10 and 10).

Assuming that the function in question is a constant function equal to zero, the parent function is f(x) = 0.

To describe the transformations that happened to this function, we need more information about the specific formula. For example, if the formula is:

g(x) = 0{-10≤x≤10}

Then there are no transformations from the parent function. The function is simply a constant function that is equal to zero over the interval [-10, 10].

However, if the formula is something like:

g(x) = 2 * 0{-10≤x+3≤10}

Then we can describe the transformations as follows:

Horizontal translation: The function has been shifted horizontally to the left by 3 units. This means that the point (3, 0) on the parent function is now located at the origin (0, 0) on the transformed function.

Dilation/reflection: The function has been reflected about the y-axis and vertically scaled by a factor of 2. This means that the point (-1, 0) on the parent function is now located at (-4, 0) on the transformed function, and the point (1, 0) on the parent function is now located at (2, 0) on the transformed function.

Vertical translation: There is no vertical translation in this case, since the constant function is already centered at y = 0.

To summarize, the formula g(x) = 2 * 0{-10≤x+3≤10} represents a function that has been horizontally shifted left by 3 units, reflected about the y-axis, and vertically scaled by a factor of 2.

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The formula g(x) = 2 * 0{-10≤x+3≤10} represents a function that has been horizontally shifted left by 3 units, reflected about the y-axis, and vertically scaled by a factor of 2.

A function is a mathematical rule that assigns each input from a set (domain) a unique output from another set (range), typically written as y = f(x).

The notation "0{-10≤x≤10}" typically represents the domain of a function or an inequality. It means that the function is defined only for the values of x that are between -10 and 10 (including -10 and 10).

Assuming that the function in question is a constant function equal to zero, the parent function is f(x) = 0.

To describe the transformations that happened to this function, we need more information about the specific formula. For example, if the formula is:

g(x) = 0{-10≤x≤10}

Then there are no transformations from the parent function. The function is simply a constant function that is equal to zero over the interval [-10, 10].

However, if the formula is something like:

g(x) = 2 * 0{-10≤x+3≤10}

Then we can describe the transformations as follows:

Horizontal translation: The function has been shifted horizontally to the left by 3 units. This means that the point (3, 0) on the parent function is now located at the origin (0, 0) on the transformed function.

Dilation/reflection: The function has been reflected about the y-axis and vertically scaled by a factor of 2. This means that the point (-1, 0) on the parent function is now located at (-4, 0) on the transformed function, and the point (1, 0) on the parent function is now located at (2, 0) on the transformed function.

Vertical translation: There is no vertical translation in this case, since the constant function is already centered at y = 0.

To summarize, the formula g(x) = 2 * 0{-10≤x+3≤10} represents a function that has been horizontally shifted left by 3 units, reflected about the y-axis, and vertically scaled by a factor of 2.

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The correct matches for the probability of falling below the z-score are:

-0.08: 0.4681

0.63: 0.7357

-2.7: 0.0035

1.95: 0.9744

Explain probability

Probability is a measure of the likelihood or chance of an event occurring. It is expressed as a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain. Probability is calculated by dividing the number of favourable outcomes by the total number of possible outcomes. Probability is used in many fields, including mathematics, statistics, science, economics, and finance, to make predictions and decisions based on uncertain events.

According to the given information

To match the probability of falling below a given z-score, we need to use a standard normal distribution table or a calculator with a built-in normal distribution function. Here are the probabilities for each z-score:

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If Riders have to climb a m staircase to board the ride at its lowest point.

a. sine function for the height of Emma, who is at the very top of the ride when t = 0 is: h(t) = 60 sin(π/6 t).

b. a cosine function for Eva, who is just boarding the ride is: h(t) = m + 60 cos(π/6 t).

c.  a sine function for Matthew is: h(t) = 30 sin(π/6 t).

(a) Let's assume that the Ferris wheel completes one full revolution in 12 minutes. The height of the Ferris wheel can be modeled by a sine function as it moves up and down periodically. When the Ferris wheel completes one revolution, it returns to its original position, so the period of the sine function is 12 minutes.

The maximum height of the Ferris wheel is 60 m, so the amplitude of the sine function is 60 m. When t = 0, Emma is at the very top of the ride, which means she is at the maximum height of the Ferris wheel. Therefore, the sine function for Emma's height, h(t), can be written as:

h(t) = 60 sin(2π/12 t)

Simplifying this equation, we get:

h(t) = 60 sin(π/6 t)

(b) Eva is just boarding the ride, which means she is at the lowest point of the ride when t = 0. The cosine function is ideal for modeling this situation, as it starts at its maximum value and reaches its minimum value after one-fourth of the period. Therefore, the cosine function for Eva's height, h(t), can be written as:

h(t) = m + 60 cos(2π/12 t)

Simplifying this equation, we get:

h(t) = m + 60 cos(π/6 t)

where m is the height of the staircase that Eva has to climb to board the ride.

(c) Matthew is at the same height as the central axle of the Ferris wheel, which means he is halfway between the maximum and minimum height of the ride. Therefore, the sine function for Matthew's height, h(t), can be written as:

h(t) = 30 sin(2π/12 t)

Simplifying this equation, we get:

h(t) = 30 sin(π/6 t)

Therefore  sine function for the height of Emma, who is at the very top of the ride when t = 0 is: h(t) = 60 sin(π/6 t).

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yes is answer   for all option   .we can check it by rules of rounding off numbers .

what is rounding ?

Rounding is the process of approximating a number to a nearby value that is easier to work with or more appropriate for a given context. When rounding, we take a number with many decimal places or significant figures and adjust it to a simpler or more convenient value with fewer decimal places or significant figures.

In the given question,

Yes, each number is rounded correctly to the nearest hundred thousand based on the rules of rounding.

To round to the nearest hundred thousand, we look at the digit in the hundred thousand place and the digit to its right (i.e., in the ten thousand place).

If the digit in the ten thousand place is 5 or greater, we round up the digit in the hundred thousand place by adding 1.

If the digit in the ten thousand place is less than 5, we leave the digit in the hundred thousand place as it is.

Using these rules, we can see that:

350000 rounded to the nearest hundred thousand is 400000 because the digit in the ten thousand place is 5, so we round up the digit in the hundred thousand place.

555555 rounded to the nearest hundred thousand is 560000 because the digit in the ten thousand place is 5, so we round up the digit in the hundred thousand place.

137998 rounded to the nearest hundred thousand is 200000 because the digit in the ten thousand place is 7, so we round up the digit in the hundred thousand place.

792314 rounded to the nearest hundred thousand is 800000 because the digit in the ten thousand place is 3, so we leave the digit in the hundred thousand place as it is.

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f(x) is concave up on the interval (-∞, √(6/7)) U (√(6/7), ∞),f(x) is concave down on the interval (-√(6/7), √(6/7)) ,The inflection points occur at x = -√(6/7) and x = √(6/7).

An inflection point is a point on a curve where the concavity changes, from concave up to concave down or vice versa, indicating a change in the curvature of the curve.

According to the given information:

To determine the intervals where f(x) is concave up or down, we need to find the second derivative of f(x) and determine its sign.

First, we find the first derivative of f(x):

f'(x) = (14x)/(7x²+6)²

Then, we find the second derivative of f(x):

f''(x) = [28(7x²+6)²- 28x(7x²+6)(4x)] / (7x²+6)^4

Simplifying the expression, we get:

f''(x) = 28(42x² - 72) / (7x^2+6)³

To determine where f(x) is concave up or down, we need to find the intervals where f''(x) is positive or negative, respectively.

Setting f''(x) = 0, we get:

42x² - 72 = 0

Solving for x, we get:

x = ±√(6/7)

These are the possible inflection points of f(x). To determine if they are inflection points, we need to check the sign of f''(x) on both sides of each point.

We can use a sign chart to determine the sign of f''(x) on each interval.

Intervals where f''(x) > 0 are where f(x) is concave up, and intervals where f''(x) < 0 are where f(x) is concave down.

Here is the sign chart for f''(x):

x  |   -∞   | -√(6/7) | √(6/7) |   ∞

f''(x)|   -    |     +      |     -     |   +

From the sign chart, we can see that:

a) f(x) is concave up on the interval (-∞, √(6/7)) U (√(6/7), ∞).

b) f(x) is concave down on the interval (-sqrt(6/7), √(6/7)).

c) The inflection points occur at x = -√(6/7) and x = √(6/7).

Therefore, the open intervals where f(x) is concave up are (-∞, -√(6/7)) and (√(6/7), ∞), and the open interval where f(x) is concave down is (-√(6/7), √(6/7)). The inflection points occur at x = -√(6/7) and x = √(6/7).

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The f(x) is concave up on the interval (-∞, √(6/7)) U (√(6/7), ∞),f(x) is concave down on the interval (-√(6/7), √(6/7)) ,The inflection points occur at x = -√(6/7) and x = √(6/7).

What is inflection Point?

An inflection point is a point on a curve where the concavity changes, from concave up to concave down or vice versa, indicating a change in the curvature of the curve.

According to the given information:

To determine the intervals where f(x) is concave up or down, we need to find the second derivative of f(x) and determine its sign.

First, we find the first derivative of f(x):

f'(x) = (14x)/(7x²+6)²

Then, we find the second derivative of f(x):

f''(x) = [28(7x²+6)²- 28x(7x²+6)(4x)] / (7x²+6)^4

Simplifying the expression, we get:

f''(x) = 28(4x² - 72) / (7x^2+6)³

To determine where f(x) is concave up or down, we need to find the intervals where f''(x) is positive or negative, respectively.

Setting f''(x) = 0, we get:

42x² - 72 = 0

Solving for x, we get:

x = ±√(6/7)

These are the possible inflection points of f(x). To determine if they are inflection points, we need to check the sign of f''(x) on both sides of each point.

We can use a sign chart to determine the sign of f''(x) on each interval.

Intervals where f''(x) > 0 are where f(x) is concave up, and intervals where f''(x) < 0 are where f(x) is concave down.

Here is the sign chart for f''(x):

x  |   -∞   | -√(6/7) | √(6/7) |   ∞

f''(x)|   -    |     +      |     -     |   +

From the sign chart, we can see that:

a) f(x) is concave up on the interval (-∞, √(6/7)) U (√(6/7), ∞).

b) f(x) is concave down on the interval (-sqrt(6/7), √(6/7)).

c) The inflection points occur at x = -√(6/7) and x = √(6/7).

Therefore, the open intervals where f(x) is concave up are (-∞, -√(6/7)) and (√(6/7), ∞), and the open interval where f(x) is concave down is (-√(6/7), √(6/7)). The inflection points occur at x = -√(6/7) and x = √(6/7).

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Therefore, the perimeter of the rectangle is 42 centimeters.

The concept of area is used in many areas of mathematics, science, and everyday life. It is used in geometry to calculate the area of various shapes, such as triangles, circles, and polygons. It is also used in physics to calculate the amount of surface area of an object that is exposed to air or water, and in architecture and engineering to determine the amount of material needed to construct a building or structure.

Here,

To find the perimeter of a rectangle, we need to know its length and width. We are given that the width of the rectangle is 9 centimeters, and the area of the rectangle is 108 square centimeters.

We can use the formula for the area of a rectangle:

Area of rectangle = length x width

Plugging in the values we have:

108cm² = length x 9cm

Solving for the length, we can divide both sides by 9cm:

length = 108cm² / 9cm

length = 12cm

So, the length of the rectangle is 12 centimeters.

To find the perimeter of the rectangle, we can use the formula:

Perimeter of rectangle = 2 x (length + width)

Plugging in the values we have:

Perimeter of rectangle = 2 x (12cm + 9cm)

Perimeter of rectangle = 2 x 21cm

Perimeter of rectangle = 42cm

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The control group for the experiment would be option C: some plants will receive no sunlight.

The control group in an experiment is the group that does not receive the treatment or intervention being tested, so that the effects of the treatment can be compared to a baseline or reference point. In this experiment, the treatment being tested is the provision of sunlight as an energy source for plant growth.

Therefore, the control group should not receive sunlight, so that the effects of sunlight can be compared to the baseline of plant growth without sunlight.

Therefore, the control group for the experiment would be option C: some plants will receive no sunlight.

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Answer:

14cm

Step-by-step explanation:

112÷8=14

base length=14cm

Answer:

To find the base length of a rectangle, given its area and height, you can use the formula for calculating the area of a rectangle, which is:

Area = Length x Width

In this case, you are given that the area is 112 square inches and the height is 8 inches. Let's denote the base length as "x" inches.

So, the equation for the area of the rectangle becomes:

112 = x * 8

To solve for "x", you can divide both sides of the equation by 8:

112 / 8 = x

x = 14

Therefore, the base length of the rectangle is 14 inches.

Answe: The distance of light bulb can be calculated using equation of parabola. The parabola is a plane curve which is U-shaped.

Step-by-step explanation:

Therefore, it will take 173 months to pay off the loan, or approximately 14 years and 5 months.

A percentage is a way of expressing a number as a fraction of 100. The symbol for a percentage is "%". For example, 50% is the same as 50/100 or 0.5 as a decimal. Percentages are often used to express a portion or share of a whole. For instance, if you scored 90% on a test, it means you got 90 out of 100 possible points. In finance, percentages are commonly used to express interest rates, returns on investments, or changes in stock prices.

First, we need to convert the monthly interest rate from a percentage to a decimal by dividing by 100.

0.25% / 100 = 0.0025

Now we can plug in the values into the formula:

N= (-log⁡ (1-0.0025*70000/250))/ (log⁡ (1+0.0025))

Simplifying the equation in the parentheses:

N= (-log⁡ (1-175))/ (log⁡ (1.0025))

N= (-log⁡ (0.9964))/ (0.002499)

N= 172.9

Rounding up to the nearest whole number since we can't make partial payments:

N= 173

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The amount in the bank after 7 years increases as the compounding frequency increases, and it is highest when interest is compounded continuously.

Using the formula A = P(1 + r/n)^(nt), where:

A = the amount in the account after t years

P = the principal (initial amount)

r = the annual interest rate (as a decimal)

n = the number of times the interest is compounded per year

t = the number of years

a) If interest is compounded annually:

A = 2000(1 + 0.05/1)^(1*7) = $2,835.08

b) If interest is compounded quarterly:

A = 2000(1 + 0.05/4)^(4*7) = $2,888.95

c) If interest is compounded monthly:

A = 2000(1 + 0.05/12)^(12*7) = $2,905.03

d) If interest is compounded continuously:

A = Pe^(rt) = 2000e^(0.05*7) = $2,938.36

Therefore, the amount in the bank after 7 years increases as the compounding frequency increases, and it is highest when interest is compounded continuously.

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The range is the best measure of variability for this data, and its value is 4.

The line plot displays the number of roses purchased per day at a grocery store, with the data values ranging from 0 to 4 (since there are no dots above 4).

The best measure of variability for this data is the range, which is the difference between the maximum and minimum values in the data set. In this case, the minimum value is 0 and the maximum value is 4, so the range is:

Range = Maximum value - Minimum value = 4 - 0 = 4

Therefore, the range is the best measure of variability for this data, and its value is 4.

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The variance of the population according to the given table is option B 41.

The spread or dispersion of a set of data around its mean is measured by variance. It has the same units as the original data and is calculated as the average of the squared deviations from the mean. Variance is a frequently used statistical term to describe the diversity or variability of a population or sample. When the variance is modest, the data points are closely grouped around the mean, whereas when the variance is great, the data points are widely dispersed.

The variance is given by the formula:

variance = (sum of squared deviations from the mean) / (number of observations)

Using the table we have sum of squared deviations from the mean:

81 + 1 + 1 + 81 = 164

variance = 164 / 4 = 41

Hence, the variance of the population according to the given table is option B 41.

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An observation is considered an outlier if it is below 35.5 or above 79.5 in this dataset.

To determine the outliers in a dataset using the five-number summary, we need to calculate the interquartile range (IQR), which is the difference between the third quartile (Q3) and the first quartile (Q1).

IQR = Q3 - Q1

Where

Q1 = 52

Q3 = 63

So, we have

IQR = 63 - 52

IQR = 11

An observation is considered an outlier if it is:

Below Q1 - 1.5 × IQR

Above Q3 + 1.5 × IQR

Substituting the values, we get:

Below 52 - 1.5 × 11 = 35.5

Above 63 + 1.5 × 11 = 79.5

Therefore, an observation is considered an outlier if it is below 35.5 or above 79.5 in this dataset.

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Thus, the total surface area of pentagonal prism is found to be 308.6 sq. ft.

A prism having a pentagonal base is referred to as a pentagonal prism. It has two hexagonal bases, five parallelogram faces, and seven faces. Seven faces, fifteen edges, and ten vertices make up a pentagonal prism.

The two bases of each of the seven faces—two pentagons—and the remaining five faces—parallelograms—connect the bases of the pentagons.

Given data:

surface area of pentagonal prism = 2* base area + 5*rectangle area

surface area of pentagonal prism = 2* B + 5*L*w

surface area of pentagonal prism = 2* 84.3 + 5*7*4

surface area of pentagonal prism = 168.6 + 140

surface area of pentagonal prism = 308.6 sq. ft

Thus, the total surface area of pentagonal prism is found to be 308.6 sq. ft.

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Therefore, the answer is (A) 7x-9 when it is given that function F(x) = 4x - 8 and g(x) = -3x + 1.

In mathematics, a function is a relationship between two sets of values, where each input (or domain element) is associated with a unique output (or range element). In other words, a function is a rule or a process that takes an input (or inputs) and produces a corresponding output. Functions can be expressed using various mathematical notations, such as algebraic formulas, graphs, tables, or even verbal descriptions. They are widely used in many fields of mathematics, science, engineering, economics, and computer science, to model and solve problems that involve relationships between variables or quantities.

Here,

To find (f - g)(x), we need to subtract g(x) from f(x), so we get:

(f - g)(x) = f(x) - g(x)

Substituting the given functions, we get:

(f - g)(x) = (4x - 8) - (-3x + 1)

Simplifying the expression by distributing the negative sign, we get:

(f - g)(x) = 4x - 8 + 3x - 1

Combining like terms, we get:

(f - g)(x) = 7x - 9

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Therefore , the solution of the given problem of probability comes out to be a)1/78 ,b)65/624 ,c)1/4 ,d)0 and e)12/13.

The basic goal of any considerations technique is to assess the probability that a statement is accurate or that a specific incident will occur. Chance can be represented by any number range between 0 and 1, where 0 normally indicates a percentage but 1 typically indicates the level of certainty. An illustration of probability displays how probable it is that a specific event will take place.

Here,

a.

P(Bert gets a Jack and Ernie rolls a five) = P(Bert gets a Jack) * P(Ernie rolls a five)

= (4/52) * (1/12)

= 1/78

b.

P(Bert gets a heart and Ernie rolls a number less than six) = P(Bert gets a heart) * P(Ernie rolls a number less than six)

= (13/52) * (5/12)

= 65/624

c.

P(Bert gets a face card and Ernie rolls an even number) = P(Bert gets a face card) * P(Ernie rolls an even number)

= (12/52) * (6/12)

= 1/4

d.

P(Bert gets a red card and Ernie rolls a fifteen) = 0

e.

Ernie rolls a number that is not twelve, and Bert draws a card that is not a Jack:

A regular 52-card deck contains 48 cards that are not Jacks,

so the likelihood that Bert will draw one of those cards is 48/52, or 12/13.

On a 12-sided dice with 11 possible outcomes,

Ernie rolls a non-12th-number (1, 2, 3, etc.).

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Therefore, it is not even chance, but it is also not very unlikely. It is moderately likely that a yellow marble will be selected on the first draw.

Probability is a branch of mathematics that deals with the study of random events and their likelihood of occurring. It is a measure of the likelihood or chance of an event happening. Probability is expressed as a number between 0 and 1, where 0 means the event is impossible, and 1 means the event is certain.

In other words, probability is a way of quantifying uncertainty. It is used in various fields, such as statistics, physics, finance, engineering, and more, to help make predictions and decisions based on uncertain information. Probability theory provides a set of rules and tools for analyzing and manipulating random variables and events, and for calculating the probability of complex events.

The likelihood of a yellow marble being selected on the first draw can be calculated by dividing the number of yellow marbles by the total number of marbles in the bag:

likelihood of selecting a yellow marble = number of yellow marbles / total number of marbles

So, in this case, the likelihood of selecting a yellow marble on the first draw is:

likelihood of selecting a yellow marble = 2 / (3 + 2 + 1) = 2/6 = 1/3

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