QUESTION #1

cost analysis with graph
office copier has an initial price of $2500. A service contract costs $200 for the first year and increases $50 per year thereafter. It can be shown that the total cost of the copier after n years is given by C(n)= 2500 +175n +25n^2. The average cost per year for n years is .

1.. Write the equation for the average cost.

2. plot the equation for the average cost for year 2 up to 20 (this is the domain of the function.

3. During what year is the average cost at a minimum (between years 2 and 20)

4. When is the average cost per year at a minimum and what is the minimum average

1. The equation for average cost per year (AC) is given by:

AC = (C(n) - C(0)) / n

2. Plotting the equation for average cost from year 2 to year 20:
Please refer to the graph below:

n | AC
---------
2 | ?
3 | ?
4 | ?
5 | ?
6 | ?
7 | ?
8 | ?
9 | ?
10 | ?
11 | ?
12 | ?
13 | ?
14 | ?
15 | ?
16 | ?
17 | ?
18 | ?
19 | ?
20 | ?

3. To determine the year when the average cost is at a minimum between years 2 and 20, we need to calculate the average cost for each year and find the smallest value.

4. To find when the average cost per year is at a minimum and what the minimum average is, we would need to calculate the average cost for each year between 2 and 20 and find the year with the lowest average cost.

1. To find the equation for the average cost, we need to divide the total cost C(n) by the number of years n. So, the equation for the average cost is:

Average cost = C(n)/n

2. To plot the equation for the average cost for years 2 up to 20, we will substitute the values of n from 2 to 20 into the equation and plot the corresponding average cost. Here is the table:

| n | C(n) | Average Cost |
|---|-------------|-------------------|
| 2 | 3450 | 1725 |
| 3 | 3900 | 1300 |
| 4 | 4400 | 1100 |
| 5 | 4950 | 990 |
| 6 | 5550 | 925 |
| 7 | 6200 | 886 |
| 8 | 6900 | 863 |
| 9 | 7650 | 850 |
|10 | 8450 | 845 |
|11 | 9300 | 845 |
|12 | 10200 | 850 |
|13 | 11150 | 857 |
|14 | 12150 | 868 |
|15 | 13200 | 880 |
|16 | 14300 | 894 |
|17 | 15450 | 909 |
|18 | 16650 | 925 |
|19 | 17900 | 942 |
|20 | 19200 | 960 |

Plotting these values on a graph will give you the graph of the average cost for years 2 to 20.

3. To find the year when the average cost is at a minimum between years 2 and 20, we can either observe the graph or find the minimum value of the average cost function. From the graph or by analyzing the table values, we can see that the average cost is at a minimum in year 10.

4. To find the minimum average cost and when it occurs, we need to find the minimum point of the average cost function. The average cost function is given by:

Average cost = 2500 + 175n + 25n^2

Taking the derivative of this function with respect to n and setting it to zero, we can find the value of n at which the average cost has a minimum. Then, substituting that value of n into the average cost function will give us the minimum average cost.

Taking the derivative, we have:

d/dn (Average cost) = 175 + 50n

Setting it to zero:

175 + 50n = 0

50n = -175

n = -3.5

Since we are dealing with the number of years, we can round the value of n to the nearest whole number, which is 4. Therefore, the average cost is at a minimum in year 4, and the minimum average cost is the value we get by substituting n=4 into the average cost function:

Average cost = 2500 + 175n + 25n^2

Putting n=4, we get:

Average cost = 2500 + 175(4) + 25(4^2)
Average cost = 2500 + 700 + 400
Average cost = 3600

To answer these questions, we need to follow step-by-step instructions and explanations. Here's how we can solve each question:

1. To find the equation for the average cost, we divide the total cost (C(n)) by the number of years (n). This gives us:
Average Cost = C(n) / n

Plugging in the expression for C(n) from the given information, we get:
Average Cost = (2500 + 175n + 25n^2) / n

Simplifying this equation further might provide a clearer representation.

2. To plot the equation for the average cost for year 2 up to 20, we need to evaluate the Average Cost equation for each year within the given domain. We substitute values of n from 2 to 20 into the equation, calculate the result for each n, and plot the points (n, Average Cost) on a graph with n on the x-axis and Average Cost on the y-axis.

3. To find the year when the average cost is at a minimum between years 2 and 20, we can use calculus. We need to find the derivative of the Average Cost equation, set it to zero, and then solve for n. The n value obtained will represent the year when the average cost is at a minimum.

4. To determine when the average cost per year is at a minimum and what the minimum average cost is, we substitute the n value obtained from the previous step into the Average Cost equation and calculate the result. This will give us both the year and the minimum average cost.

By following these steps, we can find the solutions to the questions regarding cost analysis with a graph for the office copier.