Question

The management of Ditton Industries has determined that the daily marginal revenue function associated with selling x units of their deluxe toaster ovens is given by the following, where R '(x) is measured in dollars/unit. Rtext( )'\(x\) = -0.1x + 40

(a) Find the daily total revenue realized from the sale of 210 units of the toaster oven. $
(b) Find the additional revenue realized when the production (and sales) level is increased from 210 to 310 units. $

Answer 1

(a) The daily total revenue realized from the sale of 210 units of the toaster oven is $6,195.

(b) The additional revenue realized when the production (and sales) level is increased from 210 to 310 units is $1,400.

Explanation:

The marginal revenue gives the actual revenue realized from the sale of an additional unit of the commodity given that sales are already at a certain level. The derivative R' of the function R measures the rate of change of the revenue function.

We know that the daily marginal revenue function is given by

`R'(x) = -0.1x+40`

(a) To find the the daily total revenue you must:

• Integrate the daily marginal revenue function,

`\int R'(x)dx = \int(-0.1x+40)dx\\\\R(x)=-\int \:0.1xdx+\int \:40dx\\\\R(x)=-0.05x^2+40x+C`,

where C is a constant.

• Find the value of C, using the fact that if you sell 0 units your daily revenue is $0.

`0=-0.05(0)^2+40(0)+C\\C=0`

`R(x)=-0.05x^2+40x`

The daily total revenue realized from the sale of 210 units of the toaster oven is

x = 210 units

`R(210)=-0.05(210)^2+40(210)\\R(210)=-210^2\cdot \:0.05+8400\\R(210)=-2205+8400\\R(210)=6195`

(b) To find the additional revenue realized when the production (and sales) level is increased from 210 to 310 units you must:

• Find the daily total revenue realized from the sale of 310 units

`R(310)=-0.05(310)^2+40(310)\\R(310)=-310^2\cdot \:0.05+12400\\R(310)=-4805+12400\\R(310)=7595`

The additional revenue realized when the production (and sales) level is increased from 210 to 310 units is

`R(310)-R(210)=7595-6195=1400`