Model various situations with the difference of polynomials. Simplify

Question

A bicycle company produces y bicycles at a cost represented by the polynomial y2+10y+100,000. The revenue for y bicycles is represented by 2y2+10y+500. Find a polynomial that represents the company's profit. If the company only has enough materials to make 300 bicycles, should it make the bicycles?

Answer 1

The profit function can be obtained by finding the difference of the revenue and the cost functions:

 

Let the profit function be represented by P(y). Therefore,

P(y) = 2y²+10y+500 - (y²+10y+100000)

 

P(y) = 2y2+10y+500 - y2-10y-100000

 

P(y) = y²-99500

 

If the company has enough materials to produce only 300 bicycles, then the profit or loss made will be computed as

follows:

P(300) = 300²-99500 = 90,000-99500 = -9500

 

Since this is negative, there is a loss of 9500 dollars or money units. 

 

Answer: Based on the information given, the company should not make the bicycles in order to avoid incurring losses.

 

Extension of this problem

For the company to engage in the production of the bicycles, it must possess enough materials to breakeven, that is, the number of materials to produce enough bicycles in order to have revenues equal to costs. This can be calculated by setting P(y) = 0

Thus: P(y) = y²-99500 = 0

y² = 99500

y = ±√99500 = ±315.44

However, we need to accept only the positive value of 315.44

Therefore, to engage in the production of the bicycles, the company should have enough materials to produce at least 316 bicycles, unless other conditions are negotiated.

 

Answer 2

Profit = Revenue - Cost

P =R-C

P =2y²+10y+500       -        y²+10y+100,000 = y² - 99500 

 

 Now at y=300, P = -$9500.00 which means it is not profitable to make 300 bikes.

 

NB: To figure out the number of bikes at which the company will break eve, set y2 - 99500 =0
y = 316 bicycles.