The 20 residents of the village of Towneburg are considering hosting a fireworks show next summer.The total cost of the show is estimated to be $1,000 + $10 per unit.Each resident's demand for fireworks is identical: P = 50 - 2 * units as shown below:
Suppose that Towneburg can only blast off whole numbers of fireworks units.The optimal number of units at Towneburg's fireworks display is

Question

The 20 residents of the village of Towneburg are considering hosting a fireworks show next summer.The total cost of the show is estimated to be $1,000 + $10 per unit.Each resident's demand for fireworks is identical: P = 50 - 2 * units as shown below: 
Suppose that Towneburg can only blast off whole numbers of fireworks units.The optimal number of units at Towneburg's fireworks display is

Quizplus · Accepted Answer

To find the optimal number of units, we need to calculate the total revenue generated from the fireworks show and subtract the cost of the show. The revenue is calculated by multiplying the price (P) by the number of units sold (U): R = P*U. The total revenue for Towneburg's fireworks show is:

R = (50 - 2U)*U
R = 50U - 2U^2

To find the optimal number of units, we need to maximize revenue. We can do this by taking the derivative of the revenue function and setting it equal to zero:

dR/dU = 50 - 4U
50 - 4U = 0
U = 12.5

Since we can only blast off whole numbers of fireworks units, the optimal number of units is 24 (the closest whole number to 12.5). This gives a revenue of:

R = (50 - 2*24)*24
R = $720

To calculate the profit, we subtract the cost of the show from the revenue:

Profit = R - Cost
Profit = $720 - ($1,000 + $10*24)
Profit = -$560

Since the profit is negative, it may not be a good idea for Towneburg to host a fireworks show.

The 20 Residents of the Village of Towneburg Are Considering