Question

A company manufactures and sells x VCRs per month. If the cost and revenue equations are C(x) = 200 + 2x, R(x) = 20x − (x^2/50) , use differentials to estimate the change, ∆P, in profit when the production is increased from 100 to 101.

Answer 1

ΔP = 14

Explanation:

Given

`C(x) = 200 + 2x`

`R(x) = 20x - (x^2)/(50)`

Required

Determine the change in profit

First, we need to determine the profit

`P(x) = R(x) - C(x)`

`P(x) = 20x - (x^2)/(50) - 200 - 2x`

Collect Like Terms

`P(x) = - (x^2)/(50) + 20x - 2x- 200`

`P(x) = - (x^2)/(50) + 18x- 200`

Differentiate wrt x

`P'(x) = -(x)/(25) + 18`

When Profit increases from 100 to 101, the change becomes

ΔP
`= -(x)/(25) + 18`Δx

Δx
`= (x_2 - x_1)`

Where

`x_2 = 101`

`x_1 = x = 100`

So, the expression: ΔP
`= -(x)/(25) + 18`Δx becomes

ΔP
`= (-(x)/(25) + 18)(x_2 - x_1)`

ΔP
`= (-(100)/(25) + 18)(101 - 100)`

ΔP
`= (-4+ 18)(1)`

ΔP
`= (14)(1)`

ΔP = 14