Question

the student council at greenwood high school is selling t - shirts for $20 apiece. their costs are $5 per shirt, plus $45 for supplies. selling a certain number of shirts will allow the student council to cover their costs. how many shirts do they need to sell to break even? how much in costs will they have incurred at that point? the student council needs to sell \square t - shirts to break even. at that point, they will have incurred $\square$ in costs and collected the same amount in sales.

Explanation:

Step1: Define variables and cost/sales functions

Let \( x \) be the number of T - shirts sold. The cost function \( C(x) \) includes a fixed cost of $45 for supplies and a variable cost of $5 per shirt. So, \( C(x)=5x + 45 \). The sales function \( S(x) \) is the revenue from selling T - shirts at $20 per shirt, so \( S(x)=20x \). At the break - even point, the cost equals the sales, so we set \( C(x)=S(x) \).

Step2: Solve for the number of shirts to break even

Set \( 5x + 45=20x \). Subtract \( 5x \) from both sides of the equation:
\( 45=20x - 5x \)
\( 45 = 15x \)
Then divide both sides by 15:
\( x=\frac{45}{15}=3 \)

Step3: Calculate the cost at the break - even point

Now that we know \( x = 3 \), we substitute \( x = 3 \) into the cost function \( C(x)=5x + 45 \).
\( C(3)=5\times3+45=15 + 45=60 \)

Answer:

The student council needs to sell 3 T - shirts to break even. At that point, they will have incurred $60 in costs and collected the same amount in sales.