A community theater uses the function p(d)=-4d^2+200d-100 to model the profit (in dollars) expected in a weekend when the tickets to a comedy show are priced at d dollars each. a. Write and solve an equation to find out the prices at which the theater would earn 1,500 in profit from the comedy show each weekend. Type your response in the space below. Equation: square Solutions: d = d=square or d=square b. At what price would the theater make the maximum profit? Type your answer in the box below. What is the maximum profit? Type your answer in the box below. Show your reasoning.

a. To find the prices at which the theater would earn $1,500 in profit from the comedy show each weekend, we need to set the profit function equal to $1,500 and solve for d.<br /><br />Equation: $-4d^2 + 200d - 100 = 1500$<br /><br />Solving this equation, we get:<br /><br />$-4d^2 + 200d - 1600 = 0$<br /><br />Using the quadratic formula, we find:<br /><br />$d = \frac{-200 \pm \sqrt{200^2 - 4(-4)(-1600)}}{2(-4)}$<br /><br />$d = \frac{-200 \pm \sqrt{40000 - 25600}}{-8}$<br /><br />$d = \frac{-200 \pm \sqrt{14400}}{-8}$<br /><br />$d = \frac{-200 \pm 120}{-8}$<br /><br />Solutions: $d = 25$ or $d = 35$<br /><br />b. To find the price at which the theater would make the maximum profit, we need to find the vertex of the parabola represented by the profit function.<br /><br />The x-coordinate of the vertex of a parabola in the form $y = ax^2 + bx + c$ is given by $-\frac{b}{2a}$.<br /><br />In this case, $a = -4$ and $b = 200$, so the x-coordinate of the vertex is:<br /><br />$x = -\frac{200}{2(-4)} = 25$<br /><br />Therefore, the theater would make the maximum profit when the price is $25.<br /><br />To find the maximum profit, we substitute $d = 25$ into the profit function:<br /><br />$p(25) = -4(25)^2 + 200(25) - 100$<br /><br />$p(25) = -4(625) + 5000 - 100$<br /><br />$p(25) = -2500 + 5000 - 100$<br /><br />$p(25) = 2500$<br /><br />Therefore, the maximum profit is $2,500.