Question

Each Year the Leadership Class holds a dance to raise money. The dance is rarely well attended because the sponsor belongs to an older generation that is unfamilier with today's taste in music. The profit, P, made on the dance can be modeled by the function written below. The ticket price, t, is in dollars.

P(t) = -16t² +800t - 4000 P(t) = -16(t-25)² + 6000 P(t) = -16(t² -50t +250)

1. The break even point (where profit is zero) = ?

2. The loss if the dance gets cancelled and no tickets are sold = ?

3. The number of tickets sold to maximize profit = ?

4. The maximum profit = ?

5. The number of tickets sold to make a profit of $2400 = ?

Answer 1

P(t) = -16t^2 + 800t - 4000

1) breakeven point where profit is 0

-16t^2 + 800t - 4000 = 0

-16 ( t - 25)^2 + 6000 = 0

subtrcating 6000 from both sides

-16 ( t - 25)^2 = -6000

dividing both sides by -16

( t - 25)^2 = 375

taking square root on both sides

t = 5.635 , 44.365

breakeven point is ( 5.635 , 44.365 )

2) when no tickets are sold

plugging t = 0

-16t^2 + 800t - 4000

loss = $ 4000

3) tickets to maximize profit

is the vertex

so, tickets = 25

4) maximum profit = $ 6000

5) plugging P = 2400

2400 =  -16t^2 + 800t - 4000

subtrating 2400 from both sides

-16t^2 + 800t - 6400 = 0

t = 10, 40

so tickets sold = 10, 40