A manufacture has been selling 1050 television sets a week at 360 dollars each. A market survey indicates that for each 20 -dollar rebate offered to a buyer, the number of sets sold will increase by 200 per week.
p(x)=1x/10 + 465
How large of a rebate should the company offer to a buyer, in order to maximize its revenue?
I did 697.5-360=337.5. Is this right?
nevermind I made a mistake it's p(x)=-1x/10 + 465
To determine the rebate amount that will maximize the manufacturer's revenue, we need to understand how the revenue is affected by the number of sets sold.
Let's break down the given information step by step:
1. The manufacturer is currently selling 1050 television sets a week.
2. Each television set is sold at $360, so the revenue for selling 1050 sets without a rebate is 1050 * $360 = $378,000.
Now, the market survey indicates that for each $20 rebate offered, the number of sets sold will increase by 200 per week. This means that for every additional $20 rebate, the manufacturer can sell an additional 200 sets.
To find out how much of a rebate should be offered to maximize revenue, we need to build an equation for revenue that includes the rebate amount and the number of sets sold.
Let's denote:
x = the number of $20 rebates offered
n = the additional number of sets sold per week due to the rebate
R = the total revenue
Since each rebate of $20 will increase the number of sets sold by 200, we can express n as n = 200x.
The revenue equation, R, is given as R = (1050 + n) * 360. Substituting the value of n, we get:
R = (1050 + 200x) * 360
To find the rebate amount that maximizes revenue, we need to differentiate this equation with respect to x and set it equal to zero:
dR/dx = 0
Now, let's differentiate the equation and solve for x:
dR/dx = 360 * (1050 + 200x)' = 0
dR/dx = 360 * 200 = 72000
72000 = 360 * (1050 + 200x)
Divide both sides by 360:
200 = 1050 + 200x
Subtract 1050 from both sides:
200x = 200
x = 1
The result tells us that for the revenue to be maximized, the manufacturer should offer 1 rebate of $20.
To find the actual rebate amount, multiply the number of rebates (x) by $20:
Actual Rebate Amount = x * $20 = 1 * $20 = $20
Therefore, the manufacturer should offer a $20 rebate to maximize its revenue.
Your calculation of $337.5 does not represent the correct answer. It seems you may have made an error in your calculations.