A manufacture has been selling 1400 television sets a week at $450 each. A market survey indicates that for each $14 rebate offered to a buyer, the number of sets sold will increase by 140 per week.

Question

A manufacture has been selling 1400 television sets a week at $450 each. A market survey indicates that for each $14 rebate offered to a buyer, the number of sets sold will increase by 140 per week.

a) Find the function representing the demand p(x) , where x is the number of the television sets sold per week and p(x) is the corresponding price. find 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 105000+150x , how should it set the size of the rebate to maximize its profit?

Answer 1

First part is done, look back to your previous post of this

I see you corrected your post

I suggested what to do, give it a try

Answer 2

a) To find the demand function, let's consider the given information. We know that the manufacturer sells 1400 television sets per week at a price of $450 each. We also know that for each $14 rebate offered, the number of sets sold increases by 140 per week.

Let's break down the problem to find the demand function:

At a price of $450 and without any rebate, the manufacturer sells 1400 television sets per week. So, we can write this as a point on the demand function: (1400, 450).

The rebate of $14 increases the number of sets sold by 140 per week. So, for each new price, the demand will shift to the right by 140 units.

Now we can create the demand function, which represents the price as a function of the number of sets sold:

p(x) = 450 + (x - 1400) * 14

Here, x represents the number of sets sold per week, and p(x) represents the price corresponding to that quantity.

b) To maximize revenue, we need to find the price that will result in the maximum total revenue. Total revenue is calculated by multiplying the price (p) by the quantity (x).

Revenue = p(x) * x

To find the maximum revenue, we can use calculus. First, we differentiate the revenue function with respect to x and set it equal to zero:

d(revenue)/dx = d(p(x) * x)/dx = p'(x) * x + p(x) = 0

Since we know the demand function p(x), we can differentiate it to find p'(x):

p'(x) = 14

Substituting this into the equation:

14x + p(x) = 0

Now, we can solve this equation to find x:

14x = -p(x)
14x = -450 - (x - 1400) * 14
14x = -450 - 14x + 19600
28x = 19150
x ≈ 684

So, the manufacturer should offer a rebate that results in approximately 684 television sets sold per week to maximize its revenue.

c) To maximize profit, we need to consider the cost function and revenue function. Profit is calculated by subtracting the cost from the revenue:

Profit = Revenue - Cost

Given that the weekly cost function is 105000 + 150x, we can substitute this into our profit equation:

Profit = p(x) * x - (105000 + 150x)

Now, let's substitute the demand function p(x) we found into the profit equation:

Profit = (450 + (x - 1400) * 14) * x - (105000 + 150x)

Simplifying this equation will give us the profit function as a function of x. To maximize profit, we differentiate the profit function with respect to x, set it equal to zero, and solve for x.