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 find the quantity where the marginal benefit equals the marginal cost. The marginal benefit in this case is the derivative of each resident's demand function, which is -2.

The marginal cost is a constant $10 per unit.

Setting -2 = $10, we get:

50 - 2 * units = $10
2 * units = 40
units = 20

However, we need to blast off whole numbers of units, which means we need to round up to the next highest integer, which is 24.

Therefore, the optimal number of units for Towneburg's fireworks display is 24.

The 20 Residents of the Village of Towneburg Are Considering