Question

part b: short answer
read each short answer question carefully and be sure to show all necessary work for full marks.
write each solution in the blank space provided. marks assigned to short answer questions are
indicated for each question.

1. the revenue for a production by a theatre group is ( r(t) = -50t^2 + 300t ), where ( t ) is the ticket

price in dollars. the cost for the production is ( c(t) = 600 - 50t ). determine the ticket
price(s) that will allow the production to break even. what will the revenue be at the ticket
price(s)?

Explanation:

Step1: Set break-even condition

Break-even occurs when $R(t) = C(t)$, so:
$$-50t^2 + 300t = 600 - 50t$$

Step2: Rearrange to standard quadratic form

Bring all terms to one side:
$$-50t^2 + 350t - 600 = 0$$
Divide by $-50$ to simplify:
$$t^2 - 7t + 12 = 0$$

Step3: Factor the quadratic equation

Factor the simplified quadratic:
$$(t - 3)(t - 4) = 0$$

Step4: Solve for t

Set each factor equal to 0:
$t - 3 = 0 \implies t=3$
$t - 4 = 0 \implies t=4$

Step5: Calculate revenue at t=3

Substitute $t=3$ into $R(t)$:
$$R(3) = -50(3)^2 + 300(3) = -50(9) + 900 = -450 + 900 = 450$$

Step6: Calculate revenue at t=4

Substitute $t=4$ into $R(t)$:
$$R(4) = -50(4)^2 + 300(4) = -50(16) + 1200 = -800 + 1200 = 400$$

Answer:

The break-even ticket prices are $\$3$ and $\$4$. The revenue at $\$3$ is $\$450$, and the revenue at $\$4$ is $\$400$.