Question

Question 47 ptsA community theater uses the functionp(d) = -4d? + 2000 – 100to model the profit (in dollars) expected in a weekend when the tickets to a comedy showare priced at ddollars 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. Show your reasoning.b. At what price would the theater make the maximum profit, and what is that maximum profit? Show your reasoning.

Answer 1

A community theater uses the function

p(d) = -4d^2 + 2000d – 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. Show your reasoning.



b. At what price would the theater make the maximum profit, and what is that maximum profit? Show your reasoning.

Part a)

For p(d)=$1,500

substitute in the given equation

1500=-4d^2 + 2000d – 100

solve for d

-4d^2+2000d-100-1500=0

-4d^2+2000d-1600=0

Using the formula to solve quadratic equation

we have

a=-4

b=2000

c=-1600

substitute

`d=\frac{-2000\pm\sqrt[]{(2000^2)-4(-4)(-1600)}}{2(-4)}`
`\begin{gathered} d=\frac{-2000\pm\sqrt[]{3,974,400}}{-8} \\ \\ d=(-2000\pm1,993.59)/(-8) \end{gathered}`

the solutions for d are

d=$0.8

d=$499

Part b) At what price would the theater make the maximum profit, and what is that maximum profit? Show your reasoning.

we know that

The maximum profit will be equal to the y-coordinate of the vertex of the quadratic equation

so

p(d) = -4d^2 + 2000d – 100

Find the vertex

using a graphing tool

see the attached figure

the vertex is the point (250, 249,900)

that means

the maximum profit is $249,900 and the price that would the theater make the maximum profit is $250