# Statement: A manufacturer has been selling 1000 television sets a week at <mi mathvariant="n

Statement:

A manufacturer has been selling 1000 television sets a week at $\mathrm{}480$ each. A market survey indicates that for each $\mathrm{}11$ rebate offered to a buyer, the number of sets sold will increase by 110 per week.

Questions :

a) Find the function representing the revenue R(x), where x is the number of $\mathrm{}11$ rebates offered.

For this, I got $\left(110x+1000\right)\left(480-11x\right)$. Which is marked correct

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

For this I got 17.27. Which was incorrect. I then tried 840999, the sum total revenue at optimized levels

c) If it costs the manufacturer $\mathrm{}160$ for each television set sold and there is a fixed cost of $\mathrm{}80000$, how should the manufacturer set the size of the rebate to maximize its profit?

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For part b, the question is: for what x is $R\left(x\right)=\left(110x+1000\right)\left(480-11x\right)$ maximized?

First, differentiate using the power rule and then set the derivative equal to zero:
${R}^{\prime }\left(x\right)=110\left(480-11x\right)-11\left(110x+1000\right)=-2420x+41800$
${R}^{\prime }\left(x\right)=0⇔-2420x+41800⇔x\approx 17.2$
The rebate offered must therefore be
$\mathrm{}17.2\cdot 11=\mathrm{}190\phantom{\rule{thinmathspace}{0ex}}\text{dollars}$