Answer to Question #100267 in Calculus for pei

Question

A small startup company produces lens and apertures for DSLR camera that they sell through a website. After extensive research, the company has developed a revenue function as follows

R(x,y) = x(110-4.5x) + y(155-2y)

in thousand ringgit where x is the number of lens produced and sold in thousands and y is the number of aperture produced and sold in thousands. The corresponding cost function is

C (x,y) = 3x2 + 3y2 + 5xy – 5y + 50

in thousand ringgit. If you are the investors, find the production levels that maximize the revenue. Then, would you invest or not in the company? Justify

Answer 1

$R(x,y)=x(110-4.5x) + y(155-2y)$

Now we differentiate the function partially with respect to x and y and equate it to 0, in order to find the production levels for maximum revenue generation.

$\implies \partial R/\partial x =110-9x=0$

$\implies x=110/9$

Similarly, $\partial R/\partial y=155-4y=0$

$\implies y=155/4$

Calculating revenue for these production levels, we get the maximum possible revenue, which is;

$R(110/9,155/4)=3675.347$

The cost function is given by

$C (x,y) = 3x^2 + 3y^2 + 5xy – 5y + 50$

For the same above obtained production levels, the cost is

$C(110/9,155/4)=7177.141$

Clearly, the company cannot be invested in because even for production levels generating maximum revenue, the cost of production is more than the revenue generated, thus resulting in a net loss and thus making the investment not feasible.