SOLUTION: The price p and the quantity x sold of a certain product obey the demand equation below. x= -4p+88, 0 less than or equals p less than or equals 22 ​(a) Express the revenu

Algebra ->  Finance -> SOLUTION: The price p and the quantity x sold of a certain product obey the demand equation below. x= -4p+88, 0 less than or equals p less than or equals 22 ​(a) Express the revenu      Log On


   



Question 1130303: The price p and the quantity x sold of a certain product obey the demand equation below.
x= -4p+88, 0 less than or equals p less than or equals 22
​(a)
Express the revenue R as a function of x.
​(b)
What is the revenue if 68 units are​ sold?
​(c)
What quantity x maximizes​ revenue? What is the maximum​ revenue?
​(d)
What price should the company charge to maximize​ revenue?
​(e)
What price should the company charge to earn at least ​$288 in​ revenue?

Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

The price p and the quantity x sold of a certain product obey the demand equation below.
x=+-4p%2B88, 0%3C=+p%3C=+22
p%28x%29=%2888-x%29%2F4
p%28x%29=22-x%2F4
​(a)
Express the revenue R as a function of x.
R%28x%29=p%28x%29%2Ax+
R%28x%29=+%2822-x%2F4%29x
R%28x%29=+22x-%281%2F4%29x%5E2

​(b)
What is the revenue if 68 units are​ sold?
R%2868%29=+22%2A68-%281%2F4%2968%5E2
R%2868%29=+1496-1156
R%2868%29=+340

​(c)
What quantity x maximizes​ revenue? What is the maximum​ revenue?
R%28x%29=+22x-%281%2F4%29x%5E2
R%28x%29 is a quadratic with a=-1%2F4, b=22
max occurs at x=-b%2F2a+=+%28-22%2F%28-2%2F4%29%29=+%28-22%2F%28-1%2F2%29%29=22%2A2=44
max occurs at x=44
​(d)
What price should the company charge to maximize​ revenue?
p%28x%29=22-x%2F4...max occurs at x=44
p%2844%29=22-44%2F4
p%2844%29=22-11
p%2844%29=11
the company should charge 11 to maximize​ revenue
​(e)
What price should the company charge to earn at least ​$288 in​ revenue?
288=+p%2A%28-4p%2B88%29
288=+-4p%5E2%2B88p+
72=+-p%5E2%2B22p+
p%5E2-22p%2B72=0
p%5E2-4p-18p%2B72=0
%28p%5E2-4p%29-%2818p-72%29=0
p%28p-4%29-18%28p-4%29=0
%28p-18%29%28p-4%29=0
solutions:
p=4 or
p=18