Question

A shopkeeper has 30 items to sell and want to sell each item at Rs 40 to earn some profit but for every Rs 4 increment in selling price of one item,he left with one unsold item.Find at what selling price, he would be able to maximize his profit? (Some unsold items remain with him).

Answer 1

Answer:

At Selling price of Rs. 80 he can maximize his profit to Rs. 1600.

Step-by-step explanation:

Given,

Selling Price = Rs 40 per item

Selling Price = 30 * 40 = Rs. 1200

Let say he increase 4n price

then items unsold = 4n/4 = n

Items sold = 30 - n

Price at which items sold = 40 + 4n

Selling Price = (40 + 4n)(30 - n)

                     = 4(10 + n)(30 -n)

                     = 4 ( -n² +20n + 300)

$\frac{\mathrm{d}\,SP}{\mathrm{d}n}=\frac{\mathrm{d}4(-n^2+20n+300)}{\mathrm{d}n}=4(-2n+20)$

Put,

dS/dn  = 0

4 ( -2n + 20 ) = 0

-2n + 20 = 0

=> n = 10

Price increased = 4n = Rs. 40

New Selling price = 40 + 40 = Rs 80

Items sold = 30 - 10 = 20

Maximum amount = 80 * 20 = Rs 1600

Therefore, At Selling price of Rs. 80 he can maximize his profit to Rs. 1600.