What price per cellphone unit gives them the maximum monthly sales?

Question

A company of cellular phones can sell 200 units per month at 2,000 dollars each. Then they found out that they can sell 50 more cellphone units every month for each 100 dollars decrease in price . What price per cell phone unit gives them the maximum monthly sales?

Answer 1

So we want to maximize revenue which is price times quantity.

 

R=pq

 

We also have a demand curve which determines price as a function of quantity. 

 

So the slope of the demand curve is the change in price/ change in quantity. In this case -100/50 or -2. (Negative bc the price and quantity move in opposite directions, price decreases mean quantity increases). 

 

So using the point slope form of the equation 

 

p-p₁ =-2(q-q₁). The point we sub in is 

the original numbers of (200,2000)

 

so

p-2000=-2(q-200). Simplify and solve for p

 

p-2000=-2q+400

p=-2q+2400

 

now substitute into the revenue equation

R=(-2q+2400)q

=-2q²+2400q

To maximize revenue, we can do it by setting the derivative equal to 0 or realize it is a parabola that opens down and find the vertex as -b/2a. 

 

So q=-2400/-4

q=600

so plugging into the demand equation 

gives us p=1200 

and then multiplying pq to get the maximum revenue of 720,000