Respuesta :
Answer:
(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.
Step-by-step 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
[tex]R'(x) = -0.1x+40[/tex]
(a) To find the the daily total revenue you must:
- Integrate the daily marginal revenue function,
[tex]\int R'(x)dx = \int(-0.1x+40)dx\\\\R(x)=-\int \:0.1xdx+\int \:40dx\\\\R(x)=-0.05x^2+40x+C[/tex],
where C is a constant.
- Find the value of C, using the fact that if you sell 0 units your daily revenue is $0.
[tex]0=-0.05(0)^2+40(0)+C\\C=0[/tex]
[tex]R(x)=-0.05x^2+40x[/tex]
The daily total revenue realized from the sale of 210 units of the toaster oven is
x = 210 units
[tex]R(210)=-0.05(210)^2+40(210)\\R(210)=-210^2\cdot \:0.05+8400\\R(210)=-2205+8400\\R(210)=6195[/tex]
(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
[tex]R(310)=-0.05(310)^2+40(310)\\R(310)=-310^2\cdot \:0.05+12400\\R(310)=-4805+12400\\R(310)=7595[/tex]
The additional revenue realized when the production (and sales) level is increased from 210 to 310 units is
[tex]R(310)-R(210)=7595-6195=1400[/tex]
