In the figure DB bisects

(a) The rates of changes of the volume are:

When r = 9 inches, dV/dt = 972π cubic inches per minute

When r = 36 inches, dV/dt = 15,552π cubic inches per minute

(b)  The rate of change of the volume relates to the radius through a quadratic function not a constant function.

a) We know that the formula for the volume of a sphere is V = (4/3)πr³. We can take the derivative with respect to time t to find the rate of change of volume.

dV/dt = 4πr² (dr/dt)

When r = 9 inches, we have:

dV/dt = 4π(9²)(3) = 972π cubic inches per minute

When r = 36 inches, we have:

dV/dt = 4π(36²)(3) = 15,552π cubic inches per minute

b) The rate of change of the volume of the sphere is not constant even though dr/dt is constant because the rate of change of the volume relates to the radius through a quadratic function not a constant function.

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The rule for this relation could be S = n² + 1.

What is relation rule?

A relation rule is a description of how two or more quantities or variables are related to each other. A relation rule can take various forms, depending on the nature of the relation and the variables involved.

For example, a relation rule can be expressed as an equation, a formula, a table, a graph, or a verbal description. In each case, the relation rule specifies the conditions under which the variables are related, and how they vary in relation to each other.

Based on the given table:

n: 1, 0, 2

S: 1, 0, 3

We can observe that when n=1, S=1; when n=0, S=0; and when n=2, S=3.

There are different possible rules that can describe this relation, but one possible rule is:

S = n² + 1

Using this rule, we can verify that it matches the given table:

Therefore, the rule for this relation could be S = n² + 1.

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Answer:

(16x^2 + 46x + 20)/(4x + 5) ft

Step-by-step explanation:

he area of the yard is given by (8x^2+13x+5) ft^2. The length of the yard is (8x+5) ft. The width of the yard is (8x^2+13x+5)/(8x+5) ft.

The fence is along the length and two widths of the yard. Therefore, the length of the fence is 2(8x+5) ft and the width of the fence is 2(8x^2+13x+5)/(8x+5) ft.

The total length of the fence is the sum of the length and width of the fence.

Therefore, the total length of the fence is 2(8x+5) + 2(8x^2+13x+5)/(8x+5) ft.

Simplifying the expression, we get:

2(8x+5) + 2(8x^2+13x+5)/(8x+5) = (16x^2 + 46x + 20)/(4x + 5) ft.

Therefore, Karla’s dad will need (16x^2 + 46x + 20)/(4x + 5) ft of fencing.

The probability that the sample mean will fall within $600 of the population of average family income mean is 0.3203 or 32.03%.

Population standard deviation = σ = $4800

Sample size = n = 225

Difference between the sample mean and population mean = d = $600

We need to find the probability that the sample mean will fall within $600 of the population mean.

We can use the z-score formula for sample means to find this probability:

z = (x - μ) / (σ / √n)

where z is the z-score, x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.

Substituting the given values, we get:

z = ($600) / ($4800 / √225)

z = 0.75

Using the standard normal distribution table or calculator, we can find the probability that z is between -0.75 and 0.75:

P(-0.75 < z < 0.75) = 0.5469 - 0.2266 = 0.3203

Therefore, the probability that the sample mean will fall within $600 of the population of average family income mean is 0.3203 or 32.03%.

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Answer:

#15 (a)

[tex]{ \sf{a + 2b - c + 4d}} \\ \\ \hookrightarrow \: { \sf{ \binom{4}{6} + 2 \binom{4}{ - 2} - \binom{ - 3}{ - 2} + 4 \binom{5}{1} }} \\ \\ \hookrightarrow \: { \sf{ \binom{4}{6} + \binom{8}{ - 4} + \binom{3}{2} + \binom{20}{4} \: \: \: \: \: \: }} \\ \\ { \sf{ \hookrightarrow \: \binom{4 + 8 + 3 + 20}{6 - 4 + 2 + 4} }} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ { \sf{ \hookrightarrow \: { \boxed{ \binom{45}{8} }}}}[/tex]

#15 (b)

Step-by-step explanation:

15 (a)

if it is a vector

First, we need to multiply each vector component by its corresponding scalar value and then add them together.

a + 2b - c + 4d = (4)(1) + (4)(2) + (-3)(-1) + (5)(4) = 4 + 8 + 3 + 20 = 35

Therefore, the value of a + 2b - c + 4d is 35.

if it is a matrix

we can form a 1x4 matrix for vector a, b, c, and d respectively. We can also form a 4x4 matrix for the scalar values. Then we can perform matrix multiplication as follows:

(4 4 -3 5) (1 0 0 0) (0 2 0 0) (0 0 -1 0) (0 0 0 4)

= (41 + 40 - 30 + 50) (40 + 42 - 30 + 50) (40 + 40 - 3*(-1) + 50) (40 + 40 - 30 + 5*4)

= (4, 8, 3, 20)

Therefore, the value of a + 2b - c + 4d is (4, 8, 3, 20).

15 (b)

To find the product of the two matrices, we need to multiply each element of the first matrix by its corresponding element in the second matrix and add the products. The resulting matrix will be a 2 x 2 matrix.

| 5 -2 | | 2 6 | |(5)(2) + (-2)(5) (5)(6) + (-2)(3)|

| 4 1 | x | 5 3 | = |(4)(2) + (1)(5) (4)(6) + (1)(3)|

Performing the matrix multiplication, we get:

| 5 -2 | | 2 6 | | 10 -12 |

| 4 1 | x | 5 3 | = | 13 27 |

Therefore, the product of the matrices (5 -2, 4 1) and (2 6, 5 3) is the 2 x 2 matrix:

| 10 -12 |

| 13 27 |

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1-  (2x-4)(x+5)

Multiplying using the distributive property, we get:

(2x-4)(x+5) = 2x(x) + 2x(5) - 4(x) - 4(5)

= 2x^2 + 10x - 4x - 20

= 2x^2 + 6x - 20

Therefore, (2x-4)(x+5) simplifies to 2x^2 + 6x - 20.

2-(x-2)^2

Expanding using the formula for the square of a binomial, we get:

(x-2)^2 = x^2 - 4x + 4

Therefore, (x-2)^2 simplifies to x^2 - 4x + 4.

3- (3x+1)^2

Expanding using the formula for the square of a binomial, we get:

(3x+1)^2 = (3x)^2 + 2(3x)(1) + (1)^2

= 9x^2 + 6x + 1

Therefore, (3x+1)^2 simplifies to 9x^2 + 6x + 1.

4- (3x-1)(2x^2+5x-4)

Using the distributive property, we can multiply each term in the first polynomial by each term in the second polynomial:

(3x-1)(2x^2+5x-4) = 3x(2x^2) + 3x(5x) - 3x(4) - 1(2x^2) - 1(5x) + 1(4)

= 6x^3 + 15x^2 - 12x - 2x^2 - 5x + 4

= 6x^3 + 13x^2 - 17x + 4

Therefore, (3x-1)(2x^2+5x-4) simplifies to 6x^3 + 13x^2 - 17x + 4.

Answer:

Step-by-step explanation:

x > -12

Multiplying 13 m by itself and multiplying the result by π gives us the area of the sidewalk as A = 530.94 m². We can calculate the area of the sidewalk with A = πr²

The area of the circular sidewalk around the flower bed can be calculated using the formula for the area of a circle: A = πr². To find the radius of the sidewalk, we need to subtract the radius of the flower bed from the diameter of the flower bed and the sidewalk. Therefore, the radius of the circular sidewalk is (19 m - 3 m - 3 m) = 13 m.Using the formula for the area of a circle, we can calculate the area of the sidewalk with A = πr², where r = 13 m. Substituting 13 m for r, we get A = π(13 m)². Multiplying 13 m by itself and multiplying the result by π gives us the area of the sidewalk as A = 530.94 m².

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Answer:

[tex]\left(\begin{array}{cc}0&24\\13&27\end{array}\right)[/tex]

Step-by-step explanation:

I will attempt to explain using the following example. Let us consider the  product (multiplication) of the following matrices:

[tex]\left(\begin{array}{cc}a_{1,1}&a_{1,2}\\a_{2,1}&a_{2,2}\end{array}\right) \cdot \left(\begin{array}{cc}b_{1,1}&b_{1,2}\\b_{2,1}&b_{2,2}\end{array}\right) = \left(\begin{array}{cc}c_{1,1}=c_{1\times 1}&c_{1,2}=c_{1\times2}\\c_{2,1}=c_{2\times1}&c_{1,2}=c_{2\times2}\end{array}\right)[/tex]

Note that the first number in each coefficient refers to the row number, while the second number refers to the column number.

Then [tex]a_{2,1}[/tex] indicates the element in the second row and first column.

To calculate the product of two matrices, we need to multiply each row of the first matrix by each column of the second matrix.

Example:

[tex]c_{1,1}= (a_{1,1} \cdot b_{1,1} + a_{1,2} \cdot b_{2,1})[/tex]

Now, we can apply this to the original exercise:

[tex]\left(\begin{array}{cc}5&-2\\4&1\end{array}\right) \left(\begin{array}{cc}2&6\\5&3\end{array}\right) =\left(\begin{array}{cc}c_{1,1}&c_{1,2}\\c_{2,1}&c_{2,2}\end{array}\right)[/tex]

Next, we will calculate each value of c using the multiplication process we just discussed.

[tex]c_{1,1}= (5(2)+-2(5)) = 10-10 =0\\c_{1,2}= (5(6)+-2(3)) = 30-6 =24\\c_{2,1}= (4(2)+1(5)) = 8+5 =13\\c_{2,2}= (4(6)+1(3)) = 24+3 =27\\[/tex]

Thus, we have obtained the final result of the matrix product:

[tex]\left(\begin{array}{cc}5&-2\\4&1\end{array}\right) \left(\begin{array}{cc}2&6\\5&3\end{array}\right) =\left(\begin{array}{cc}0&24\\13&27\end{array}\right)[/tex]

[tex]\text{-B$\mathfrak{randon}$VN}[/tex]

The probability of reaching into the bag and choosing either a blue or yellow piece of candy is 38/60 or 56%.

What is probability?

The probability of an occurrence is a figure that represents how likely it is that the event will take place. In terms of percentage notation, it is expressed as a number between 0 and 1, or between 0% and 100%. The higher the likelihood, the more likely it is that the event will take place.

Here, we have

Given: a bag has 10 blue,12 green,8 yellow,11 pink, and 10 orange pieces of candy.

The probability of reaching into the bag and choosing a blue piece of candy is 5/10 or 50%.

The probability of reaching into the bag and choosing a yellow piece of candy is 3/5 or 60%.

Hence, the probability of reaching into the bag and choosing either a blue or yellow piece of candy is 38/60 or 56%.

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Answer:

  132 in

Step-by-step explanation:

You want to know the perimeter of a triangle after the dimensions are quadrupled.

"Quadrupled" means "multiplied by 4."

The perimeter is the sum of the lengths of the sides of the triangle:

  P = 8 +10 +15 = 33 . . . . inches

If the sides are quadrupled, it is ...

  P' = 4(8) +4(10) +4(15) = 4(8 +10 +15) = 4(33) = 132 . . . . inches

As you can see, if the sides are quadrupled, the perimeter is quadrupled.

The new perimeter is 132 inches.

Answer:

A.

Step-by-step explanation:

Because If you are simplifying then -3 would become -1 and -5 would become -3 and -4 would become -2 so your basically just simplifying

The metro bus travels 5 miles per gallon of gas.

The formula used to calculate the distance traveled per gallon of gas is number of miles traveled divided by number of gallons used. Therefore, to calculate how far the metro bus travels in one gallon, we use the following formula:

Distance per gallon = Miles Traveled / Gallons Used

Distance per gallon = 275 miles / 55 gallons

Distance per gallon = 5 miles

Therefore, the metro bus travels 5 miles per gallon of gas.

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Here's the tree diagram for the given scenario:

(1) The probability of getting tails and a 3 is the probability of following the path T-3, which is 1/2 x 1/4 = 1/8.

(ii) The probability of getting heads and an even number is the probability of following the path H-2 or H-4, which is 1/2 x 1/2 + 1/2 x 1/2 = 1/2.

(iii) The probability of not getting heads and an even number is the probability of following the paths T-2 or T-4, which is 1/2 x 1/2 + 1/2 x 1/2 = 1/2.

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Answer:

$89.04

Step-by-step explanation:

The frequency for boys is 16% and the frequency for girls is 19%. Adding these two values together gives us the marginal relative frequency of 16%.

The marginal relative frequency of students who prefer swimming as their preferred type of exercise is 16%. Marginal relative frequency is a measure of the proportion of a certain group in relation to the total population. In this case, the marginal relative frequency of students who prefer swimming as their preferred type of exercise is 16%, which means that out of the total population of students, 16% prefer swimming as their preferred type of exercise.To calculate this, we need to add up the frequency of swimming for boys and girls. The frequency for boys is 16% and the frequency for girls is 19%. Adding these two values together gives us the marginal relative frequency of 16%.

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The perimeter of the shape is approximately 314.

The perimeter of a shape made from part of a circle can be found using the formula P = 2πr + 2s, where r is the radius of the circle and s is the length of the straight line segments. For example, if a shape is composed of a semicircle with radius 5 and two straight line segments of length 10, then the perimeter can be calculated as P = 2π(5) + 2(10) = 30π + 20 = 100π. To solve this calculation, we can use the value of π, which is approximately 3.14, to get the approximate perimeter of the shape as 100π ≈ 314.

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Complete question

What is the perimeter of a shape made from part of a circle composed of a semicircle with radius 5 and two straight line segments of length 10?

The probability of having an error when confidence limits are 3 standard deviations far from the mean is approximately 0.27.

Assume that the data is normally distributed. For a normal distribution, approximately 99.7% of the data is within 3 standard deviations of the mean.

As a result, if the lower and upper specification limits are 3 standard deviations away from the mean, they encompass 99.7% of the data.

As a result, there is only a 0.3% chance of an error. Since this is a two-tailed test, the probability of an error on each side is 0.15%

As a result, the probability of an error when the lower and upper specification limits are 3 standard deviations away from the mean is approximately 0.27.

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The FR of the hydraulic jack is 1.94, the VR is 1.95, and the Efficiency is 90%.

To find the FR, VR, and Efficiency of the hydraulic jack, we can use the following formulas:

FR = (F2 / F1)

VR = (D1 / D2)

Efficiency = (F2 x D2) / (F1 x D1)

where F1 is the input force, D1 is the input distance, F2 is the output force, and D2 is the output distance.

Given:

F1 = 6800N

D1 = 0.76m

F2 = ?

D2 = 0.39m

m = 1162kg

To find F2, we can use the formula for work:

Work input = Work output

F1 x D1 = F2 x D2

(6800N) x (0.76m) = F2 x (0.39m)

F2 = (6800N x 0.76m) / 0.39m

F2 = 13,216.41N

To find VR, we can use the formula:

VR = (D1 / D2) = (0.76m / 0.39m) = 1.95

To find Efficiency, we can use the formula:

Efficiency = (F2 x D2) / (F1 x D1)

Efficiency = (13,216.41N x 0.39m) / (6800N x 0.76m)

Efficiency = 0.90 or 90%

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According to the given information the limit of the given equation is 1.25 so the correct answer is option a.

Limit, a closeness-based mathematical notion, is largely used to give values to some functions at locations where none are specified, in a manner that is compatible with neighbouring values.

The value that a function assumes when its input approaches as well as approaches a particular number is really the function's limit. Limits determine continuity, integrals, as well as derivatives. The behaviour of the function at a certain place is always of relevance to the limit of the function.

[tex]\lim_{x \to 3} \frac{x^2-x-6}{x^2 - 2x - 3} \\\\ \lim_{x \to3} \frac{(x-3)(x+2)}{(x-3)(x+1)} \\\\ \lim_{x \to3} \frac{(x+2)}{(x+1)} \\\\$putting the value x=3$\\\\=\frac{3+2}{3+1} \\\\=\frac{5}{4} \\\\=1.25[/tex]

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The following random variable meets the criteria for a hypergeometric distribution: Out of 50 adults, 10 who have a graduate degree. A sample of 20 is taken. Define x to be the number of adults in the sample with a graduate degree.

A hypergeometric distribution is a type of probability distribution that measures the likelihood of a particular number of successes (in this case, adults with a graduate degree) in a sample without replacement.

A random variable that meets the criteria for a hypergeometric distribution is that which meets these conditions: A sample of size n is taken from a population of size N. The population has k successes and N - k failures.

The hypergeometric distribution is used when sampling is done without replacement, where the sample size is small relative to the population size. The sample size is typically less than 10% of the population size. When you have the information of the population, you can use the hypergeometric distribution.

Out of the options provided, the random variable that meets the criteria for a hypergeometric distribution is, Out of 50 adults, 10 who have a graduate degree. A sample of 20 is taken. Define x to be the number of adults in the sample with a graduate degree.

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The probability of a home for sale having a garage but no pool is 0.42.

To do this, we need to use some probability rules. One useful rule is the formula for calculating the probability of an event not occurring. This is given by:

P(not A) = 1 - P(A)

where A is the event we are interested in, and not A is the event of A not occurring.

In this case, we are interested in the probability of a home having a garage but no pool. Let's call this event GNP (short for garage no pool). Using the information given, we know that 60% of homes have garages and 18% have both garages and pools. We can use this information to calculate the probability of a home having a garage but no pool as follows:

P(GNP) = P(G) - P(G and P)

where G is the event of a home having a garage and P is the event of a home having a pool.

Substituting the values we have:

P(GNP) = 0.6 - 0.18

P(GNP) = 0.42

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Comparing the estimated probabilities with the actual probability, we can see that Keeley had the best estimate for the probability.

a) To find out who had the best estimate for the probability of the spinner landing on 2, we need to compare their estimated probabilities with the actual results. The actual probability of the spinner landing on 2 can be found by dividing the total number of times it landed on 2 by the total number of spins.

Sam's estimated probability = 21/80 = 0.2625
Keeley's estimated probability = 20/50 = 0.4
Cody's estimated probability = 23/70 = 0.3286

The actual probability of the spinner landing on 2 = (21+20+23)/(80+50+70) = 0.315



b) To work out the estimated probability of the spinner landing on 2 by combining all of their results, we need to add up the total number of times the spinner landed on 2 and divide it by the total number of spins.

Total number of times the spinner landed on 2 = 21 + 20 + 23 = 64
Total number of spins = 80 + 50 + 70 = 200

Estimated probability of the spinner landing on 2 = 64/200 = 0.32

c) Using the combined results will give a better estimate than using only one person's results. This is because combining the results gives a larger sample size, which makes the estimate more accurate and reliable.

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The cost of constructing the path at the rate of Rs. 25 per m2 would be Rs. 38,465.

To find the cost of constructing the path, we need to determine the area of the path and then multiply it by the cost per square meter.

The total diameter of the pond and the path is 21m + 3.5m + 3.5m = 28m.

So the radius of the pond is half of the diameter, which is 21m/2 = 10.5m.

The radius of the pond with the path is (21m+3.5m)/2 = 12.25m.

Area of the path = Area of the outer circle - Area of the pond

[tex]= π(12.25)^2 - π(10.5)^2[/tex]

[tex]= π[(12.25)^2 - (10.5)^2][/tex]

= 3.14 x (24.5 + 10.5) x (24.5 - 10.5)

= 3.14 x 35 x 14

= 1538.6 m2 (rounded to one decimal place)

Therefore, the cost of constructing the path at the rate of Rs. 25 per m2 would be:

Cost = Area x Rate per m2

= 1538.6 x 25

= Rs. 38,465.

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The y-intercept is 5.5 inches, which is the initial height of the plant before it started growing.

We can use the slope-intercept form of a linear equation, y = mx + b, where m is the slope and b is the y-intercept, to solve the problem.

Since the plant is growing at a constant rate of 1.4 inches per day, the slope of the line that represents its growth is 1.4.

Let y be the height of the plant after x days. We know that the plant was 5.5 inches tall on day 0, so we have the point (0, 5.5) on the line.

Using the point-slope form of a linear equation, we get:

y - 5.5 = 1.4x

Simplifying, we get:

y = 1.4x + 5.5

Comparing the equation to the slope-intercept form, we see that the y-intercept is 5.5.

Therefore, the y-intercept is 5.5 inches.

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Step-by-step explanation:

let the number be y

one-third of the number = 1/3 × y = y/3

the total is 3054

therefore,. y/3 = 3054

multiply both sides by 3

y/3 × 3 = 3054 × 3

y = 9162

Thus, the whole number is 9162

3054 is 1/3 of a whole number.

In order to find the whole number, you need to find the other 2/3's. Since we already have the 1/3, we can easily solve this problem.

3054x3 = 9162

3054+3054+3054 = 9162

The whole number is 9162.

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[tex](5x^8y^4)/(15x^3y^8) = (x^5/y^4) * (1/3)[/tex] is the fully simplified expression using only positive exponents.

To simplify [tex](5x^8y^4)/(15x^3y^8)[/tex], we can start by dividing the numerator and denominator by their greatest common factor, which is[tex]5x^3*y^4[/tex]:

[tex](5x^8y^4)/(15x^3y^8) = (x^8/x^3) * (y^4/y^8) * (1/3)[/tex]

Now we can simplify each term separately. First, [tex]x^8[/tex] divided by [tex]x^3[/tex] equals [tex]x^5[/tex], because when we divide two terms with the same base, we subtract their exponents:

[tex]x^8/x^3 = x^(8-3) = x^5[/tex]

Similarly, [tex]y^4[/tex]divided by[tex]y^8[/tex] equals [tex]1/y^4[/tex], because when we divide two terms with the same base, we subtract their exponents and change the sign:

[tex]y^4/y^8 = y^(4-8) = 1/y^4[/tex]

Finally, 1/3 is already in its simplest form. Putting it all together, we get:

[tex](5x^8y^4)/(15x^3y^8) = (x^5/y^4) * (1/3)[/tex]

This is the fully simplified expression using only positive exponents.

We can state that the original equation represents a fraction, where the numerator is the product of 15, the third power of x, and the eighth power of y, and the denominator is the product of 5, the eighth power of x, and the fourth power of y.

We then determined the numerator and denominator's largest common factor, which is 5x*3*y*4, in order to condense the formula. Using the exponentiation principles, we divided the numerator and denominator by this factor before simplifying each term. The end result is a fraction with x raised to the fifth power as the numerator, y raised to the fourth power as the denominator, and 3 as the common factor.

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The x-coordinates range from -5 to 25, and the y-coordinates range from -4 to 5.

To choose appropriate scales for the coordinate plane, first, observe the range of x and y coordinates for the given points J(20, 3), K(25, 3), L(15, -3), M(-5, 5), and N(-5, -4).
The x-coordinates range from -5 to 25, and the y-coordinates range from -4 to 5.
For the x-axis, we can use a scale of 5 units for each grid square.

This scale will allow us to cover the entire range of x-coordinates.

The grid will have 6 squares in the positive x-direction and 2 squares in the negative x-direction.
For the y-axis, we can use a scale of 1 unit for each grid square.

This scale will allow us to cover the entire range of y-coordinates.

The grid will have 5 squares in the positive y-direction and 4 squares in the negative y-direction.
In summary, use a scale of 5 units for each grid square on the x-axis, and a scale of 1 unit for each grid square on the y-axis.

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To obtain the remaining funds necessary to purchase the skateboard, Seth must labor for 13 hours at a rate of $7.25 per hour.

To find out how many hours Seth needs to work to earn the remaining money, we need to first calculate how much money he still needs to earn:

Money needed = Cost of skateboard - Money in the bank

Money needed = $167 - $72.75

Money needed = $94.25

Now we can use Seth's hourly rate to figure out how many hours he needs to work to earn the remaining money:

Hours needed to work = Money needed / Hourly rate

Hours needed to work = $94.25 / $7.25 per hour

Hours needed to work = 13 hours (rounded up to the nearest hour)

Therefore, Seth needs to work for 13 hours at $7.25 per hour to earn the remaining money needed to buy the skateboard.

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