Question

A carpenter wants to sell 40 chairs. If he sells them
at * 156 per chair, he would be able to sell all the
chairs. But for every * 6 increase in price, he will
be 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 :

Total numbers of chairs = 40

If he sells them at $ 156 per chair, he would be able to sell all the  chairs

And

For every $6 increase in price, he will  be left with one additional unsold chair

To Find :

The selling price which would he be able to maximize his profits

Solution :

If he sells them at $ 156 per chair, he would be able to sell all the  chairs

For every $6 increase in price, he will  be left 1 chair

So,

Let Number of chair should sell = 40 - x

And

Let Price per chair should be = 156 + 6 x

∴ Total price of chair = Number of chair should sell × Price per chair

                                  = ( 40 - x ) × ( 156 + 6 x )

                                  = 46240 + 240 x - 156 x - 6 x²

                                  = 46240 + 84 x -  6 x²

Now, For maximum profit

Let  y = 46240 + 84 x -  6 x²

i.e   $\dfrac{dy}{dx}$ = 0

Or,   $\dfrac{d (46240 + 84 x - 6x^{2} ) }{dx}$  = 0

Or,  $\dfrac{d 46240}{dx}$ +  $\dfrac{d 84 x}{dx}$ - $\dfrac{d6x^{2} }{dx}$ = 0

Or,  0 + 84 - 6 × 2 x = 0

Or,     84 - 12 x = 0.

Or,  12 x = 84

 ∴       x = $\dfrac{84}{12}$

i.e      x = 7

Again

∵  Price per chair should be = 156 + 6 x

So, put the value of x

i.e   Price per chair should be = 156 + 6 × 7

                                                = $ 156 + $ 42

                                                = $ 198

Hence, The Price per chair should be to maximize carpenter profits is $ 198   Answer