https://file.aiquickdraw.com/mathbot/file/1728253091443m1k63epy.jpg

Please answer the question, and give only the answer

(a) The revenue function $ R(x) = 55x $, since each sweatshirt sells for $55.

(b) To find the total cost function $ C(x) $, we use the given cost information: the total cost is $45,534 for 100 sweatshirts and $63,540 for 240 sweatshirts. 

Using the two points $(100, 45534)$ and $(240, 63540)$, we can calculate the slope (variable cost per sweatshirt):
\[
m = \frac{63540 - 45534}{240 - 100} = \frac{18006}{140} = 128.61
\]
So, the variable cost per sweatshirt is approximately $128.61.

Now we calculate the fixed cost using the total cost equation for either point:
\[
C(x) = 128.61x + b
\]
Using $ x = 100 $ and total cost $ C(100) = 45534 $:
\[
45534 = 128.61(100) + b \implies b = 45534 - 12861 = 32673
\]
Thus, the total cost function is $ C(x) = 128.61x + 32673 $.

(c) To find the break-even quantity, we set $ R(x) = C(x) $:
\[
55x = 128.61x + 32673
\]
Solving for $ x $:
\[
55x - 128.61x = 32673 \implies -73.61x = 32673 \implies x = \frac{32673}{73.61} \approx 444 \text{ sweatshirts}.
\]

A company distributes college logo sweatshirts and sells them for $55 each. The total cost function is linear, and the total cost for 100 sweatshirts is $45,534, whereas the total cost for 240 sweatshirts is $63,540. Write the equation for the revenue function, the total cost function, and find the break-even quantity.

Solve a problem involving a linear cost function, revenue function, and break-even analysis for a company selling sweatshirts. Given the total cost for 100 and 240 sweatshirts, calculate the break-even quantity. Suitable for high school students in Grades 10-12.

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