Question

A school uses the function p(t)=t2−20t+175 to model the profit (in dollars) expected in a weekend when the tickets are sold at a certain price, t, to the school musical go on sale a. Write an equation to find out the prices at which the school would earn $100 in profit from the school musical each weekend. b. Find out at what prices the tickets should be sold at to make a $100 profit each weekend. SHOW ALL WORK.

Answer 1

`(a)\ t^2 - 20t + 75= 0`

`(b)`
`t = 5; t -=15`

Explanation:

Given

`p(t) = t^2 - 20t + 175`

Solving (a): Equation when profit = 100.

This implies that:

`p(t) =100`

So, we have:

`100 = t^2 - 20t + 175`

Rewrite as:

`t^2 - 20t + 175 - 100 = 0`

`t^2 - 20t + 75= 0`

Solving (b): Calculate t, in (a)

In a, we have:

`t^2 - 20t + 75= 0`

Expand

`t^2 - 15t -5t + 75= 0`

Factorize

`t(t - 15) -5(t - 15)= 0`

Factor out t - 15

`(t - 5)(t - 15)= 0`

Split

`t - 5 = 0; t - 15 =0`

Solve

`t = 5; t -=15`

The prices are $5 and $15, respectively