Question

A carpenter wants to sell 40 chairs. If he sells themat * 156 per chair, he would be able to sell all thechairs. But for every 6 increase in price, he willbe left with one additional unsold chair. At what sell-ing price would he be able to maximise his profits(assuming unsold chairs remain with him)?(a) 198(b) 192(c) 204(d) 210​

Answer 1

Given:

$\text{If he sells at \ 156 per chair, he sells all 40 chairs}$

$\text{Every \ 6 increase in price, he sells 1 less}$

To Find:

$\text{Selling price per chair to maximise his profit}$

$\boxed {\textbf{Solution}}$

Form the equation:

$\text{Number of chairs he should sell} = 40 - x$

$\text{Price per chair he should sell} = 156 + 6x$

$\text{Revenue} = (40 - x)(156 + 6x)$

$\text{Revenue} = 46240 + 240x - 156x - 6x^2$

$\text{Revenue} = 46240 + 84x - 6x^2$

Find x when the profit is maximum:

$y= 46240 + 84x - 6x^2$

$\dfrac{dy}{dx} = 84 - 12x$

$\text{When } \dfrac{dy}{dx} = 0,$

$84 - 12x = 0$

$-12x = -84$

$x = 7$

Find the best selling price:

$\text{Best Price Per Chair} = 156 + 6(x)$

$\text{Best Price Per Chair} = 156 + 6(7)$

$\text{Best Price Per Chair} = \ \:198$

$\boxed{\boxed { \textbf{Answer: Best Price for his chair is (a) 198}}}$