Question

A utensil manufacturer produces ‘x’ dinner sets per week and sells each set at ₹ p, where `x = (600 - p)/8`. The cost of production of ‘x’ sets is ₹ x² + 78x + 2000.

1. Write the revenue function.   [1]
2. Write the profit function.   [1]
3. Calculate the number of dinner sets to be produced and sold per week to ensure maximum profit.   [2]

Answer 1

Given, `x = (600 - p)/8`

600 – p = 8x

p = 600 – 8x

a. Revenue function

R = x × p

R(x) = x(600 – 8x)

R(x) = 600x – 8x²

b. Given cost function

C(x) = x² + 78x + 2000

Profit (P) = Revenue (R) – Cost (C)

= (600x – 8x²) – (x² + 78x + 2000)

P(x) = –9x² + 522x – 2000

c. For maximum profit

P(x) = –9x² + 522x – 2000

Differentiate w.r.t. x, we get

P’(x) = –18x + 522

P’(x) = 0

–18x + 522 = 0

⇒ `x = 522/18`

x = 29

Now, Again differentiate w.r.t. x

P’(x) = –18 < 0

The profit function has a maximum value at 29.

Number of sets produced = 29 sets.