Question

You can now sell 50 cups of lemonade per week at 39¢ per cup , but demand is dropping at a rate of 5 cups per week each week. Assuming that raising the price does not affect demand , how fast do you have to raise your price if you want to keep your weekly revenue constant.

Answer 1

3.9¢ per cup per week

Explanation:

Revenue is the product of price and quantity sold. The rate of change in any of these quantities can be found by solving the derivative equation at using the given values.

Relation of rates of change

Revenue (r) is the product of price (p) and quantity sold (q):

r = pq

Differentiating with respect to time, we have ...

r' = p'q +pq'

Given values

The values given for the problem are ...

• r' = 0
• p' = rate to be found
• p = 39 (cents per cup)
• q' = -5 (cups per week)
• q = 50

Using the above derivative relation, we have ...

0 = p'(50) +39(-5) . . . . . . fill in given values

Solution

p' = 195/50 = 3.9 . . . . . . add 195, divide by 50

The price must be raised at the rate of 3.9 cents per week to maintain unchanging revenue.