quadratic function

Question

A manufacture has been selling 1200 television sets a week at $390 each. A market survey indicates that for each $29 rebate offered to a buyer, the number of sets sold will increase by 290 per week.

a) Find the demand function p(x)�(�), where x� is the number of the television sets sold per week.

p(x)=�(�)=   

b) How large rebate should the company offer to a buyer, in order to maximize its revenue?

$  

c) If the weekly cost function is 78000+130x78000+130�, how should it set the size of the rebate to maximize its profit?

Answer 1

a.

The demand function here is the relationship between price and units sold (sometimes the demand function is represented inversely as units sold as a function of price, hence the name).

We have the following relationships between price and rebates given, and units sold and rebates given (which we denote with k):

p = standard price - rebate amount * number of rebates given = $390 - $29 * k

Now, since each rebate given corresponds to 290 extra sales, we get the final formula:

p = $390 - $29 * x / 290 = $390 - $0.10 * x

b. In order to maximize revenue, which is defined as R = p(x) * x (or: price per unit times number of units sold), we find the value at which the derivative of the revenue wrt x is zero:

R' (x) = 0

This is found as follows:

R'(x) = (p * x)' = (390 * x - 0.10*x^2)' = 390 - 0.2 * x

The derivative of the revenue vanishes for x = 390 / 0.2 = 1950.

Using the demand function to obtain the price, we get

p(1950) = $390 - $0.10 * 1950 = $195

This means a rebate of $390 - $195 = $195 results in the optimal revenue.

c. Here, your question is hard to understand due to images not rendering properly. Would you be able to type out the formulas explicitly? For example using the convention that "^" means "to the power of", e.g. X^2 meaning "x squared".