SOLUTION: The relationship between the number of units sold of a certain product, x, and the price obtained, p, is: x = 100-2p a) Find an expression for the total revenue for selling

Algebra ->  Customizable Word Problem Solvers  -> Finance -> SOLUTION: The relationship between the number of units sold of a certain product, x, and the price obtained, p, is: x = 100-2p a) Find an expression for the total revenue for selling       Log On

Ad: Over 600 Algebra Word Problems at edhelper.com


    

Question 1028198: The relationship between the number of units sold of a certain product, x, and the price
obtained, p, is:
x = 100-2p
a) Find an expression for the total revenue for selling x units of the product.
b) How many units must be sold for the marginal revenue to be 25?
The total cost of producing x units of the product is:
TC (x)=0.5x^2+10x+100
c) Find an expression for the profit when producing and selling x units of the product.
d) How many units must be produced and sold to obtain maximum profit?
e) What is the maximum profit?
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
x = 100 - 2p ==> p+=+%28100+-+x%29%2F2
a) Revenue = R%28x%29+=+xp+=+%28x%28100+-+x%29%29%2F2+=+-x%5E2%2F2+%2B50x
b) Marginal revenue is R'(x) = -x + 50.
==> -x + 50 = 25 ==> -x = -25 ==> x = 25 units must be sold for the marginal revenue to be 25.
c) Profit =+R%28x%29+-+TC%28x%29+=+-x%5E2%2F2+%2B50x+-+x%5E2%2F2-10x+-+100+=+-x%5E2+%2B+40x+-100+.

d) Maximum number of units is obtained from x+=+-b%2F%282a%29+=+-40%2F%282%2A-1%29+=+20 units.
e) maximum profit = p%2820%29+=+-20%5E2+%2B40%2A20+-+100+=+-400+%2B800+-+100+=+300.
Thus the maximum profit is $300.