item sold per week for $. Decrease price and more items sold. How many items to sell to maximize revenue

Question

A skateboard shop sells about 50 skateboards per week for $70.  For each $1 decrease in price, about 1 more skateboard per week is sold.  How many skateboards does the shop need to sell to maximize revenue weekly?

Answer 1

Answer: 60 skateboards

Quantity Demanded is inversely related to price

Q = 50 when P =70

Q = 51 when P=69

Q=0 when P =120

Q=120 when P=0

Slope = -1 Use the point slope formula y=mx+b or Q=-P+b = 50= -70+120 b=120

Q=-P+120

is the equation for the demand curve

Revenue = PQ = P(-P+120) = -P² + 120P

Take the derivative of the Revenue function

R' = -2P + 120 and set it equal to zero and solve for P

-2P+120 = 0

2P=120

P = $60

That is the revenue maximizing price

plug it into the demand curve equation Q=-P+120

Q=-60+120 = 60

the number of skateboards sold that maximizes revenue is 60 at a price of $60, giving maximum revenue of PQ = R = 60x60 = $3,600 for a week