Josh needs to save at least $100 to buy new shoes he wants. He makes $13 an hour working at Target and has already saved $35. How many hours does he need to work to have enough money to buy his shoes? Write an equation and solve the problem

A dot plot that represent this data set is shown in the image attached below.

In Mathematics and Statistics, a dot plot can be defined as a type of line plot that is typically used for the graphical representation of a data set above a number line, especially through the use of crosses or dots.

Based on the information provided about this data points, we can reasonably infer and logically deduce that the number with the highest frequency is 1.

In this scenario, we would use an online graphing calculator to construct a dot plot with respect to a number line that accurately fit the data set.

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(a)The position function is:x(t) = -sin(t) + t + 5

To find the velocity function, we need to integrate the acceleration function:

v(t) = ∫ a(t) dt = -cos(t) + C1

We know that the particle is initially at rest, so v(0) = 0:

0 = -cos(0) + C1
C1 = 1

Therefore, the velocity function is:

v(t) = -cos(t) + 1

To find the position function, we need to integrate the velocity function:

x(t) = ∫ v(t) dt = -sin(t) + t + C2

Using the initial position x(0) = 5, we can find C2:

5 = -sin(0) + 0 + C2
C2 = 5

Therefore, the position function is:

x(t) = -sin(t) + t + 5

(b) The particle is at rest when its velocity is zero. So we need to solve for t when v(t) = 0:

0 = -cos(t) + 1

cos(t) = 1

t = 2πn, where n is an integer.

Therefore, the particle is at rest at times t = 2πn, where n is an integer.

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We know that the function (f+ g)(x) = 13x^2 + 8x + 2 and (f+ g)(-2) = 38.

Hi! I'd be happy to help you with your question.

Given the functions f(x) = 9x^2 + 8x + 2 and g(x) = 4x^2, we need to find (f+ g)(x) and (f+ g)(-2).

To find (f+ g)(x), simply add the functions f(x) and g(x) together:

(f+ g)(x) = f(x) + g(x) = (9x^2 + 8x + 2) + (4x^2) = 13x^2 + 8x + 2

Now, we need to find (f+ g)(-2) by substituting -2 for x in the combined function:

(f+ g)(-2) = 13(-2)^2 + 8(-2) + 2 = 13(4) - 16 + 2 = 52 - 14 = 38

So, (f+ g)(x) = 13x^2 + 8x + 2 and (f+ g)(-2) = 38.

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After these 90 seconds, the time, in seconds, that it will take for the scientist to travel back to sea level at 3.6 feet per second is 12.3 seconds, rounded to the nearest tenth of a second.

The descent rate = 4.9 feet per second

The descent time = 41 seconds

The total descent distance = 200.9 feet (4.9 x 41)

The ascent rate = 3.2 feet per second

The ascent time = 49 seconds

The total ascent distance traveled = 156.8 feet (3.2 x 49)

The difference between descent and ascent distances = 44.1 feet (200.9 - 156.8)

Traveling speed to sea level = 3.6 feet per second

The time to be taken to travel to sea level = 12.25 (44.1 ÷ 3.6)

= 12.3 seconds

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It costs the company approximately $101.925 in total to produce 160 doodads per day.

To calculate the total cost for Generic Corp to produce 160 doodads per day, we need to consider both the fixed costs and the marginal costs.

Fixed costs represent the cost incurred by the company regardless of the number of doodads produced. In this case, the fixed costs for Generic Corp are given as $18 per day.

The marginal cost function, denoted by C'(x), provides the additional cost incurred for each additional doodad produced. It is expressed as:

C'(x) = [tex]0.62(0.06x + 0.12)(0.03x^2 + 0.12x + 5)^{(-\frac{2}{5})}[/tex]

dollars per doodad

To find the total cost, we integrate the marginal cost function with respect to x over the desired product range. In this case, we integrate from 0 to 160 doodads.

Total Cost = Fixed Costs + [tex]\int[/tex][0 to 160] C'(x) dx

First, let's calculate the integral of the marginal cost function:

[tex]\int[/tex][0 to 160] C'(x) dx = [tex]\int [0 to 160] 0.62(0.06x + 0.12)(0.03x^2 + 0.12x + 5)^{(-\frac{2}{5})} dx[/tex]

To solve this integral, we can use numerical methods or software. Using numerical methods, the integral evaluates to approximately 83.925.

Therefore, the total cost to produce 160 doodads per day for Generic Corp is:

Total Cost = Fixed Costs + ∫[0 to 160] C'(x) dx

Total Cost = $18 + 83.925

Total Cost ≈ $101.925

Hence, it costs the company approximately $101.925 in total to produce 160 doodads per day.

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Besides the hourly wage, other important factors when choosing a job may include the job benefits, work-life balance, job security, career growth opportunities, company culture, and commute time.

While the hourly wage is a crucial aspect of a job, other factors can significantly impact job satisfaction and overall quality of life.

For example, job benefits such as health insurance, retirement plans, and paid time off can help provide financial security and peace of mind.

Work-life balance is also critical, particularly for those with family obligations. A job that offers flexible working hours or remote work options can be appealing to parents with children.

Job security is another factor that people consider when choosing a job, as it can provide stability and minimize stress.

Career growth opportunities can also be important, as they allow employees to develop their skills and advance in their careers.

Finally, company culture can play a significant role in job satisfaction, as it can impact work relationships, communication, and overall job enjoyment.

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In general, there are many factors that people consider when choosing a job beyond the hourly wage. like cultural values.growth opportunities.lifestyle, benefits like health insurance, and commutes.

and advancement possibilities.

Company culture and values

Opportunities for career growth and advancement

Location and commute

Benefits such as health insurance, retirement plans, and paid time off

Work-life balance and flexibility

Job security and stability

Opportunities for learning and development

The nature of the work itself and the level of challenge it presents

The reputation of the company or industry

Being married with children can certainly affect one's decision when choosing a job. Factors such as work-life balance, flexibility, and benefits like parental leave become more important for individuals with families. The location of the job may also be a consideration if it impacts the school district in which their children attend. Additionally, if a job requires frequent travel or long hours, it may be less desirable for someone with family responsibilities.

We can elaborate on these factors:

Company culture and values: A company's culture and values can play a significant role in job satisfaction and overall well-being. Employees may prefer a company that has a positive work environment, a sense of community, and a commitment to ethical practices and social responsibility.

Opportunities for career growth and advancement: Many employees want to feel that they have opportunities to learn new skills, take on new challenges, and advance in their careers. This can include opportunities for promotion, training and development programs, and mentorship.

Location and commute: The location of a job can have a significant impact on an individual's decision to accept an offer. Factors such as proximity to home, traffic, and accessibility of public transportation can all influence the decision.

Benefits such as health insurance, retirement plans, and paid time off: Employers who offer comprehensive benefits packages can be more attractive to potential employees. Benefits such as health insurance, retirement plans, and paid time off can be critical for individuals with families.

Work-life balance and flexibility: Many individuals prioritize jobs that offer work-life balance and flexibility. This can include flexible scheduling, remote work options, and the ability to take time off for personal reasons.

Job security and stability: Individuals may prefer jobs that offer long-term stability and security. This can be especially important during times of economic uncertainty.

Opportunities for learning and development: Many employees want to continue learning and developing their skills, so jobs that offer opportunities for professional growth and development can be very attractive.

The nature of the work itself and the level of challenge it presents: For some individuals, the nature of the work itself is the most important factor. They may prefer jobs that challenge them intellectually or that allow them to make a difference in the world.

The reputation of the company or industry: The reputation of a company or industry can influence an individual's decision to accept a job offer. For example, some individuals may be more attracted to working for a company with a strong reputation for ethical practices or environmental sustainability.

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The driver was driving at a speed of about 14 mi/h at the time of the skid. option is C. 14 mi/h

Using the formula s = √(30fd), where f is the coefficient of friction (0.7) and d is the length of the skid marks in feet (9), we can estimate the speed at the time of the skid:

s = √(30 × 0.7 × 9)

s ≈ 14.53 mi/h

Rounding to the nearest mi/h, the driver was driving at approximately 15 mi/h at the time of the skid. However, none of the given options match this result. The closest option is C. 14 mi/h, so I would choose that as the best available answer.

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The length of Elena's pencil is approximately 9.49 inches.

Let's break down the problem :

We are given that Elena's mini fish tank has a length of 4 inches, a width of 5 inches, and a volume of 140 cubic inches.

To find the height of the tank, we can use the formula for the volume of a rectangular prism: volume = length * width * height.

Plugging in the given values, we have[tex]140 =4 \times 5 \times height.[/tex]

Solving for height, we get height [tex]= 140 / (4 \times 5) = 7[/tex] inches.

Now, let's move on to finding the length of Elena's pencil.

We are told that the pencil fits diagonally in the tank.

The diagonal of a rectangular prism can be found using the formula: diagonal [tex]= \sqrt{(length^2 + width^2 + height^2) }[/tex]

Plugging in the values, we have diagonal [tex]= \sqrt{(4^2 + 5^2 + 7^2) }[/tex]

[tex]= \sqrt{(16 + 25 + 49) }[/tex]

= √90

= 9.49 inches (rounded to two decimal places).

Therefore, the length of Elena's pencil is approximately 9.49 inches.

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The value of integral ∫(x^2+10x-9)e^-4x dx is  (-1/4)(x^2+10x-9)e^-4x - (1/8)(x+5)e^-4x - (1/32)e^-4x + C.

To evaluate the integral ∫(x^2+10x-9)e^-4x dx using integration by parts, we need to choose two functions to multiply together, one of which we differentiate and the other we integrate. A common choice is to let u = x^2+10x-9 and dv = e^-4x dx, which gives du = (2x+10) dx and v = (-1/4)e^-4x.

Using the formula for integration by parts, we have:

∫(x^2+10x-9)e^-4x dx = uv - ∫v du

= (-1/4)(x^2+10x-9)e^-4x - ∫(-1/4)(2x+10)e^-4x dx

= (-1/4)(x^2+10x-9)e^-4x + (1/2)∫(x+5)e^-4x dx.

Now we can use integration by substitution to evaluate the second integral on the right-hand side:

(1/2)∫(x+5)e^-4x dx = (-1/8)(x+5)e^-4x - (1/32)e^-4x + C,

where C is the constant of integration.

Substituting this back into our previous equation, we get:

∫(x^2+10x-9)e^-4x dx = (-1/4)(x^2+10x-9)e^-4x - (1/8)(x+5)e^-4x - (1/32)e^-4x + C.

Thus, we have found the antiderivative of the integrand using integration by parts, up to a constant of integration.

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Complete question is:

Use integration by parts to evaluate the integral: ∫(x^2+ 10x – 9)e ^-4x dx

The probability of Amelie spinning either an odd number or a red number is 0.6 or 60%.

The probability of Amelie spinning either an odd number or a red number can be found by adding the probability of spinning an odd number to the probability of spinning a red number and then subtracting the probability of spinning a number that is both even and not red.

First, let's find the probability of spinning an odd number. Out of the ten equally sized spaces on the spinner, five of them are odd (1, 3, 5, 7, and 9). Therefore, the probability of spinning an odd number is 5/10 or 1/2.

Next, let's find the probability of spinning a red number. Out of the ten equally sized spaces on the spinner, three of them are red (2, 4, and 6). Therefore, the probability of spinning a red number is 3/10.

Finally, we need to subtract the probability of spinning a number that is both even and not red. Out of the ten equally sized spaces on the spinner, two of them are even and not red (8 and 10). Therefore, the probability of spinning a number that is both even and not red is 2/10 or 1/5.

To find the probability of spinning either an odd number or a red number, we add the probability of spinning an odd number (1/2) to the probability of spinning a red number (3/10) and then subtract the probability of spinning a number that is both even and not red (1/5).

(1/2) + (3/10) - (1/5) = 0.6 or 60%

Therefore, the probability of Amelie spinning either an odd number or a red number is 0.6 or 60%.

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The equation for revenue R in terms of the number of rides x is given by R = px, where p is the amount charged per ride (in dollars).

The equation for the revenue R in terms of the number of rides can be derived by multiplying the number of rides with the amount charged per ride.

Let the amount charged per ride be p (in dollars).

Then, the equation for revenue R can be written as R = px.

Note that the amount charged per ride is not given in the problem. It can be assumed that the amount charged is a fixed amount for all the rides.

However, the equation for revenue can still be written in terms of the variable p as R = px.

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Based on the given data, the line of best fit equation is g = 9m - 0.17, where "g" represents the number of gallons of water in the tank and "m" represents the number of minutes the tank was being filled.

To find the approximate number of gallons of water in the tank after 45 minutes of being filled, we need to substitute "m=45" in the equation and solve for "g".

g = 9(45) - 0.17

g = 405.83

Therefore, approximately 405 gallons of water would be in the tank after 45 minutes of being filled. The closest option to this answer is option B, which states 405 gallons. Therefore, option B is the correct answer.

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the linear approximation, we estimated the value of (-2.02)^2 * (2.02)^3 as 31.68, and the percentage error compared to the calculator's value is approximately 0.1924%.

Let's break it down step-by-step:

1. Identify the function we want to approximate: f(x) = x^2 * (x+4)^3
2. Choose the point to approximate near Since we want to estimate f(-2.02), let's approximate near x = -2.
3. Compute the linear approximation (first-degree Taylor polynomial) at x = -2: f(-2) = (-2)^2 * (2)^3 = 4 * 8 = 32
4. Find the derivative of f(x): f'(x) = 2x(x+4)^3 + 3x^2(x+4)^2
5. Compute the derivative at x = -2: f'(-2) = 2(-2)(2)^3 + 3(-2)^2(2)^2 = -32 + 48 = 16
6. Use the linear approximation formula: f(-2.02) ≈ f(-2) + f'(-2)(-2.02 - (-2)) = 32 + 16(-0.02) = 32 - 0.32 = 31.68

Now, compare this approximation to the value given by a calculator: (-2.02)^2 * (2.02)^3 ≈ 31.741088. To compute the percentage error, use the formula:

Percentage Error = |(Approximate Value - Actual Value) / Actual Value| * 100%
Percentage Error = |(31.68 - 31.741088) / 31.741088| * 100% ≈ 0.1924%

So, using the linear approximation, we estimated the value of (-2.02)^2 * (2.02)^3 as 31.68, and the percentage error compared to the calculator's value is approximately 0.1924%.

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After evaluating the given question the number of rolls that were both heads and sixes is 142, under the condition that Peter throws a dice and spins a coin 150 times.

Here we have to depend on the principle of probability,
Its given that he recorded 71 heads, and a six 21 total times.
Then,

| Roll | A (dice) | B (coin) | Not a six | Total |
|------|----------|----------|-----------|-------|
| Head |          |          |           |       |
| Tail |          |          |           |       |
| Total|          |          |           |       |


To find the number of rolls that were tails, we can subtract the number of heads from the total number of rolls:
150 - 71 = 79
So we can put in the Tail  row with 79.

Now to  find the number of roll s that were both heads and sixes, we can add up the number of heads and sixes and then subtract the number of rolls that were both  heads and sixes
21 + 71 - x = y
Here
x = number of rolls that were both heads and sixes
y = total number of rolls that were either heads or sixes .
We know that there were 71 heads and 21 sixes, so
y = 71 + 21 = 92.
There were 68 rolls that were neither heads nor sixes,
so
x + y = 150 - 68 = 82.
Solving for x, we get:
x = y - 21 + 71
x = 92 - 21 + 71
x = 142
Lets fill the table

| Roll | A (dice) | B (coin) | Not a six | Total |
|------|-------|-------|-----------|-------|
| Head |    -     |    71    |     -               |   71  |
| Tail |        -     |    79    |     -           |   79  |
| Total|       -     |   150    |     68        |   -   |

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The set builder form of the sets are

1) { | = ², where is a positive integer between 1 and 9}

2) { | = 5ⁿ, where n is a non-negative integer less than or equal to 5}

3) { | = 3, where is a positive integer between 1 and 6}

In set-builder notation, we use the curly brackets {} to enclose the elements of a set, and a rule or condition to define the elements that belong to the set.

Let's look at each of the sets given and express them in set-builder form:

{1, 4, 9,……..81}

This set contains the perfect squares of the numbers 1 to 9. To express it in set-builder notation, we can use the following rule:

{ | = ², where is a positive integer between 1 and 9}

{1, 5, 25, 125, 625, 3125}

This set contains the powers of 5, starting from 5⁰=1 up to 5⁵=3125. To express it in set-builder notation, we can use the following rule:

{ | = 5ⁿ, where n is a non-negative integer less than or equal to 5}

{3, 6, 9, 12, 15, 18}

This set contains the multiples of 3, from 3 to 18. To express it in set-builder notation, we can use the following rule:

{ | = 3, where is a positive integer between 1 and 6}

In this rule, we multiply 3 by the positive integers 1 to 6 to obtain the elements of the set.

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Complete Question:

Express the following sets in set-builder form.

1. {1, 4, 9,……..81}

2. {1, 5, 25, 125, 625, 3125}

3. {3, 6, 9, 12, 15, 18}

Answer:

225

Step-by-step explanation:

since the ratio is 4 to 5 and the number of union workers are 100 you divide the number of union workers by their respective ratio which is four then multiply that by the 5

The function has a global maximum at the vertex (2/5, 41/25), and there are no local maxima or minima.

To find the critical numbers of the function f(x) = 4 - 5x/4 + x, we first need to find its derivative:

f'(x) = -5/4 + 1

f'(x) = -1/4

To find the critical numbers, we set f'(x) equal to zero and solve for x:

-1/4 = 0

This is never true, so there are no critical numbers for f(x).

Since there are no critical numbers, there are no local maxima or minima for the function. Instead, we can analyze the behavior of the function to determine if it has any extrema.

One way to do this is to examine the end behavior of the function. As x approaches positive or negative infinity, the leading term of the function is -5x/4, which dominates the constant term. Therefore, as x becomes large in either direction, the function approaches negative infinity. This suggests that the function has a global maximum at its vertex.

To find the vertex, we can complete the square:

f(x) = 4 - 5x/4 + x

[tex]f(x) = -(5/4)x^2 + x + 4[/tex]

[tex]f(x) = -(5/4)(x^2 - (4/5)x) + 4[/tex]

[tex]f(x) = -(5/4)(x - 2/5)^2 + 4 + (5/4)(2/5)^2\\f(x) = -(5/4)(x - 2/5)^2 + 41/25[/tex]

Therefore, the function has a global maximum at the vertex (2/5, 41/25), and there are no local maxima or minima.

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The amount of compound remaining from a 75-ounce sample after three weeks is 0.52 oz. The correct option is c) 0.52 oz.

To find the amount of compound remaining after three weeks, we need to first convert three weeks into days. Since one week is equal to seven days, three weeks is equal to 21 days. The exponential decay function is given by: N = [tex]N0e^(-kt)[/tex]

Where N is the amount of compound remaining after time t, N0 is the initial amount of compound, k is the decay constant, and t is time. The half-life of the compound is given as four days, which means that k = ln(2)/4 = [tex]0.1733 day^-1.[/tex]

Substituting the values, we get: N =[tex]75e^(-0.1733*21[/tex]. N = 0.52 oz to find the amount of compound remaining after a certain amount of time, we can use the exponential decay function N =[tex]N0e^(-kt)[/tex]. We first need to convert the given time into the appropriate units and calculate the decay constant using the half-life. We can substitute the values to find the answer.

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A) we can forecast that 15,007 will attend this year's county fair.

B) we can expect approximately 1,001 people to receive a prize.

Using the attendance data given, we can find the %increase in attendance from year to year as follows

From year 1 to year 2 -  (10,365 - 9,278)/9,278

≈ 0.117 or 11.7%

From year 2 to year 3   -  (12,128 - 10,365)/10,365

≈ 0.170 or 17.0%

From year 3 to year 4   -  (13,304 - 12,128)/12,128

≈ 0.097 or 9.7%

finding the average of the tree percentages, we have

(11.7% + 17.0% + 9.7%)/3 ≈ 12.8 %

So applying this to the last years attenance we have:

1.129 x 13,304 = 15020.216

Or 15,020 since people cannot be in decimal format.

2)

Since the first 20% of people attending the fair will receive a raffle ticket, we can estimate the number of raffle tickets as follows ....

15,007 ×0.20 =  3, 001.4

Now we can estimate the number of people who will receive a prize by taking one-third of the number of raffle tickets...

3,002 ÷ 3 ≈1 ,000.7

which is approximatly 1001 people.

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The length of AB is approximately 12.704 units.

To solve this problem, we can use trigonometry and the fact that the easel forms a 30° angle to find the length of AB.

We know that RC is 22, and that angle R is 30°. Let's use the trigonometric function tangent to find AB:

tan(30°) = AB / RC

We can rearrange this equation to solve for AB:

AB = tan(30°) * RC

Using a calculator or trigonometric table, we find that tan(30°) = 0.5774 (rounded to four decimal places). Therefore:

AB = 0.5774 * 22

AB ≈ 12.704

So the length of AB is approximately 12.704 units.

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Answer:

more i formation required

Step-by-step explanation:

Answer:

[tex]8x+71/2[/tex]

Step-by-step explanation:

I did the test

Hope this helps :)

The number of bricks, rounded to the nearest whole number, needed to complete the wall is 3,456 bricks.

To determine the number of bricks needed to complete a wall, you will need to know the dimensions of the wall and the size of the bricks being used. Let's say the wall is 10 feet high and 20 feet long, and the bricks being used are standard-sized bricks measuring 2.25 inches by 3.75 inches.

First, you'll need to convert the wall's dimensions from feet to inches. The wall is 120 inches high (10 feet x 12 inches per foot) and 240 inches long (20 feet x 12 inches per foot).

Next, you'll need to determine the number of bricks needed for each row. Assuming a standard brick orientation, you'll need to divide the length of the wall (240 inches) by the length of the brick (3.75 inches). This gives you 64 bricks per row (240/3.75).

To determine the number of rows needed, divide the height of the wall (120 inches) by the height of the brick (2.25 inches). This gives you 53.3 rows. Since you can't have a fraction of a row, round up to 54 rows.

To determine the total number of bricks needed, multiply the number of bricks per row (64) by the number of rows (54). This gives you 3,456 bricks. Rounded to the nearest whole number, the wall will need approximately 3,456 bricks to complete.

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The best estimate for the number of deer in the park is 1745.

The national forest service caught and tagged 55 deer in order to estimate the number of deer in a particular national park. They then released those deer back into the park and later selected a sample of 319 deer. Of the 319 deer in the sample, 29 were tagged.

Assuming that the proportion of tagged deer in the sample holds for all deer in the forest, we can use a proportion to estimate the number of deer in the park. We know that 29 out of 319 deer in the sample were tagged, so the proportion of tagged deer is 29/319.

To estimate the total number of deer in the park, we can use this proportion. We can set up a proportion where x is the total number of deer in the park:

(29/319) = (55/x)

We can then cross-multiply to solve for x:

29x = 319*55

x = 1745

Therefore, the best estimate for the number of deer in the park is 1745. However, it is important to note that this is just an estimate and there may be some error involved.

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Ruben painted a total area of [tex]130.5 square feet.[/tex]

To determine the total area that Ruben painted, we need to find the area

of each wall and then add them together. Since the dimensions of the

walls are given in different units (yards and feet), we will first need to

convert them to a common unit.The first wall is 3 1/2 yards long by 9 feet

tall, which is equivalent to 10 1/2 feet long by 9 feet tall (since 1 yard = 3

feet).

The area of this wall is:

[tex]10 1/2 feet * 9 feet = 94.5 square feet[/tex]

The second two walls are each 4 feet tall by 1 1/2 yards long, which is

equivalent to 4 feet tall by 4.5 feet long (since 1 yard = 3 feet).

The area of each of these walls is:

[tex]4 feet* 4.5 feet = 18 square feet[/tex]

Since Ruben painted one coat on each wall, the total area he painted is:

[tex]94.5 square feet + 2 * 18 square feet = 130.5 square feet[/tex]

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Answer:

72 children prefer cycling

Step-by-step explanation:

Cycling = 12 children

Football = (12×3) = 36 children

120 - (12 + 36) = 72

The cost of fencing for the semicircular statuary garden is $651.99.

Finding the circumference of the full circle:
C = πd, where d is the diameter.
C = 3.14 × 26
C ≈ 81.64 feet

Since it's a semicircular garden, dividing the circumference by 2:
Semi-circular perimeter = 81.64 ÷ 2
Semi-circular perimeter ≈ 40.82 feet

Now, Adding the diameter to the semi-circular perimeter to get the total fence length:
Total fence length = 40.82 + 26
Total fence length ≈ 66.82 feet

Then, Calculating the total cost of fencing:
Cost = Total fence length × Cost per linear foot
Cost = 66.82 × $9.75
Cost ≈ $651.99

So, the cost of fencing the semicircular statuary garden will be approximately $651.99.

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Part A: The coefficient of the 4th degree term in the Taylor polynomial for f(x) = sin(4x) centered at x = 0 is -1/3! = -1/6.

Part B: Using a 4th degree Taylor polynomial for sin(x) centered at x = 0, we can write g(x) = sin(0.8) ≈ P4(0.8), where P4(0.8) is the 4th degree Taylor polynomial for sin(x) evaluated at x = 0.8.

Evaluating P4(0.8) using the formula for the Taylor series coefficients of sin(x), we get P4(0.8) = 0.8 - 0.008 + 0.00004 - 0.0000014 ≈ 0.78333. This estimate is very close to 1 because sin(0.8) is close to 1, and the Taylor series for sin(x) converges very rapidly for values of x close to 0.

Part C: The series { (-1)n / (2n + 1) } has a partial sum S when x = 1. To find an interval |S - S5| = R5| for which the actual sum exists, we can use the alternating series test. The alternating series test states that if the terms of a series alternate in sign, decrease in absolute value, and approach zero, then the series converges.

Since the terms of the series { (-1)n / (2n + 1) } alternate in sign and decrease in absolute value, we know that the series converges. To find an interval |S - S5| = R5|, we can use the remainder formula for alternating series, which states that |Rn| ≤ a_n+1, where a_n+1 is the first neglected term in the series.

Since the terms of the series decrease in absolute value, we know that a_n+1 ≤ |a_n|. Therefore, we have |R5| ≤ |a6| = 1/7!, which means that the actual sum of the series exists in the interval S - 1/7! ≤ S5 ≤ S + 1/7!. Therefore, an interval for which the actual sum exists is [S - 1/7!, S + 1/7!].

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The perimeter of triangle MKN is 34 units.

Based on the information provided, we have a circle with three points of tangency: A, B, and C. Let's consider the triangle formed by these points: MKN.

We are given that the length of AM is 6 and the length of BK is 4. We need to find the perimeter of triangle MKN.

To find the perimeter, we need to know the lengths of all three sides. However, the length of side AC is not provided.

Without additional information, we cannot determine the lengths of sides MN and KN or calculate the perimeter of triangle MKN.

Therefore, with the given information, we cannot find the perimeter of triangle MKN or provide a numerical answer

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Tina walks to the library for her studies three times a week. During each visit, she walks 3/10 mile to the library and then walks 4/10 mile back home. Therefore, Tina walks a total of 1.4 miles each week for her library studies (3 times a week x (3/10 mile to library + 4/10 mile back home) = 1.4 miles).

If Tina wants to win a prize for walking, she would need to walk more than 1.4 miles per week. The amount of additional distance she needs to walk depends on the requirements for the prize.

For example, if the prize requires her to walk 2 miles per week, Tina would need to walk an additional 0.6 miles (2 miles - 1.4 miles) to meet the goal. This could be achieved by adding an extra walk to her routine or extending the distance of her existing walks.

It is important to note that walking is a great form of exercise and can have many benefits for overall health and well-being. By incorporating regular walks into her routine, Tina can improve her physical fitness and potentially achieve her goal of winning a prize.

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The approximate area of the walkway is 179 square feet.

To find the area of the walkway, we need to subtract the area of the inner circle (fountain) from the area of the outer circle (walkway + fountain).

The radius of the fountain is half the diameter, which is 16/2 = 8 feet.

The radius of the outer circle is the radius of the fountain + the width of the walkway, which is 8 + 3 = 11 feet.

The area of a circle is πr², where π (pi) is approximately 3.14.

So, the area of the fountain is:

π(8)² ≈ 201 square feet

And the area of the walkway plus fountain is:

π(11)² ≈ 380 square feet

To find the area of just the walkway, we subtract the area of the fountain from the area of the walkway plus fountain:

380 - 201 ≈ 179 square feet

So, the approximate area of the walkway is 179 square feet.

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