Answer :
Certainly! Let's break down the solution step-by-step.
### Part (A)
To find the average cost per unit when 500 frames are produced, we use the formula for the average cost per unit, which is the total cost divided by the number of units produced.
Given the total cost function:
[tex]\[ C(x) = 70,000 + 400x \][/tex]
For [tex]\( x = 500 \)[/tex] frames, the total cost is:
[tex]\[ C(500) = 70,000 + 400 \times 500 \][/tex]
Now we calculate the total cost:
[tex]\[ C(500) = 70,000 + 200,000 = 270,000 \][/tex]
The average cost per unit is:
[tex]\[ \bar{C}(500) = \frac{C(500)}{500} = \frac{270,000}{500} \][/tex]
Calculating the division gives:
[tex]\[ \bar{C}(500) = 540 \][/tex]
So, the average cost per unit when 500 frames are produced is [tex]\(\$540\)[/tex].
### Part (B)
To find the average value of the cost function over the interval [tex]\([0, 500]\)[/tex], we use the formula for the average value of a function over an interval [tex]\([a, b]\)[/tex], given by:
[tex]\[ \bar{C} = \frac{1}{b-a} \int_a^b C(x) \, dx \][/tex]
In this case, [tex]\(a = 0\)[/tex], [tex]\(b = 500\)[/tex], and [tex]\(C(x) = 70,000 + 400x\)[/tex].
[tex]\[ \bar{C} = \frac{1}{500 - 0} \int_0^{500} (70,000 + 400x) \, dx \][/tex]
To solve this, we first find the integral:
[tex]\[ \int (70,000 + 400x) \, dx \][/tex]
[tex]\[ = 70,000x + \frac{400x^2}{2} \][/tex]
[tex]\[ = 70,000x + 200x^2 \][/tex]
We evaluate this from [tex]\(0\)[/tex] to [tex]\(500\)[/tex]:
[tex]\[ \left[ 70,000x + 200x^2 \right]_0^{500} = (70,000(500) + 200(500^2)) - (70,000(0) + 200(0^2)) \][/tex]
[tex]\[ = (70,000 \times 500 + 200 \times 250,000) \][/tex]
[tex]\[ = 35,000,000 + 50,000,000 \][/tex]
[tex]\[ = 85,000,000 \][/tex]
Now we divide by the length of the interval [tex]\(b - a = 500 - 0 = 500\)[/tex]:
[tex]\[ \bar{C} = \frac{85,000,000}{500} \][/tex]
[tex]\[ = 170,000 \][/tex]
Next, we find the average value by considering the average of the costs over each unit interval and dividing by the number of units:
[tex]\[ = \frac{170,000}{500} \][/tex]
[tex]\[ = 340 \][/tex]
So, the average value of the cost function over the interval [tex]\([0, 500]\)[/tex] is [tex]\(\$340\)[/tex].
### Part (C)
The difference between parts (A) and (B) lies in what we are averaging:
- In part (A), we are finding the average cost per unit when exactly 500 units are produced. This average takes into account the fixed cost (70,000) and the variable cost per unit (400) for 500 units.
- In part (B), we are finding the average value of the cost function over the interval [tex]\([0, 500]\)[/tex]. This is essentially averaging the cost function over every single unit produced from 0 to 500, including the fixed cost and how it is spread out over smaller quantities and larger quantities.
The average cost per unit when 500 frames are produced (\[tex]$540) is higher than the average value of the cost function over the interval \([0, 500]\) (\$[/tex]340). This difference occurs because part (A) considers the fixed cost fully allocated over 500 units, while part (B) averages the total cost over all units produced from 0 to 500, effectively distributing the fixed cost more broadly and incorporating lower costs when fewer units are produced.
### Part (A)
To find the average cost per unit when 500 frames are produced, we use the formula for the average cost per unit, which is the total cost divided by the number of units produced.
Given the total cost function:
[tex]\[ C(x) = 70,000 + 400x \][/tex]
For [tex]\( x = 500 \)[/tex] frames, the total cost is:
[tex]\[ C(500) = 70,000 + 400 \times 500 \][/tex]
Now we calculate the total cost:
[tex]\[ C(500) = 70,000 + 200,000 = 270,000 \][/tex]
The average cost per unit is:
[tex]\[ \bar{C}(500) = \frac{C(500)}{500} = \frac{270,000}{500} \][/tex]
Calculating the division gives:
[tex]\[ \bar{C}(500) = 540 \][/tex]
So, the average cost per unit when 500 frames are produced is [tex]\(\$540\)[/tex].
### Part (B)
To find the average value of the cost function over the interval [tex]\([0, 500]\)[/tex], we use the formula for the average value of a function over an interval [tex]\([a, b]\)[/tex], given by:
[tex]\[ \bar{C} = \frac{1}{b-a} \int_a^b C(x) \, dx \][/tex]
In this case, [tex]\(a = 0\)[/tex], [tex]\(b = 500\)[/tex], and [tex]\(C(x) = 70,000 + 400x\)[/tex].
[tex]\[ \bar{C} = \frac{1}{500 - 0} \int_0^{500} (70,000 + 400x) \, dx \][/tex]
To solve this, we first find the integral:
[tex]\[ \int (70,000 + 400x) \, dx \][/tex]
[tex]\[ = 70,000x + \frac{400x^2}{2} \][/tex]
[tex]\[ = 70,000x + 200x^2 \][/tex]
We evaluate this from [tex]\(0\)[/tex] to [tex]\(500\)[/tex]:
[tex]\[ \left[ 70,000x + 200x^2 \right]_0^{500} = (70,000(500) + 200(500^2)) - (70,000(0) + 200(0^2)) \][/tex]
[tex]\[ = (70,000 \times 500 + 200 \times 250,000) \][/tex]
[tex]\[ = 35,000,000 + 50,000,000 \][/tex]
[tex]\[ = 85,000,000 \][/tex]
Now we divide by the length of the interval [tex]\(b - a = 500 - 0 = 500\)[/tex]:
[tex]\[ \bar{C} = \frac{85,000,000}{500} \][/tex]
[tex]\[ = 170,000 \][/tex]
Next, we find the average value by considering the average of the costs over each unit interval and dividing by the number of units:
[tex]\[ = \frac{170,000}{500} \][/tex]
[tex]\[ = 340 \][/tex]
So, the average value of the cost function over the interval [tex]\([0, 500]\)[/tex] is [tex]\(\$340\)[/tex].
### Part (C)
The difference between parts (A) and (B) lies in what we are averaging:
- In part (A), we are finding the average cost per unit when exactly 500 units are produced. This average takes into account the fixed cost (70,000) and the variable cost per unit (400) for 500 units.
- In part (B), we are finding the average value of the cost function over the interval [tex]\([0, 500]\)[/tex]. This is essentially averaging the cost function over every single unit produced from 0 to 500, including the fixed cost and how it is spread out over smaller quantities and larger quantities.
The average cost per unit when 500 frames are produced (\[tex]$540) is higher than the average value of the cost function over the interval \([0, 500]\) (\$[/tex]340). This difference occurs because part (A) considers the fixed cost fully allocated over 500 units, while part (B) averages the total cost over all units produced from 0 to 500, effectively distributing the fixed cost more broadly and incorporating lower costs when fewer units are produced.
