Question

A company produces two varieties of pens: alpha and beta. Each alpha pen needs twice as much labour time as the beta pen. If only beta pens are manufactured, then the company can produce 500 pens per day. The market can take only up to 150 alpha pens and 250 beta pens per day from the company. The profits earned by the company on selling an alpha pen and a beta pen are rs 8 and rs 5 respectively. What are the respective numbers of alpha and beta pens that the company needs to produce in order to maximize the profit?

Answer 1

Answer:

Therefore, number of alpha pens produced = 125 and number of beta pens produced=250.

Step-by-step explanation:

Let x be the number of alpha pens produced.

Let y be the number of beta pens produced.

It is given that each alpha pen needs twice as much labor time as the beta pen, and the company can produce 500 beta pens per day.

Therefore, the company can produce 250 alpha pens per day.

Hence, each alpha pen can be produced in x/250 part of the day, and each beta pen can be produced in y/500 part of the day.

Hence, $\frac{x}{250}+\frac{y}{500}=1\\  2x+y=500\:\:\: (1)$

It is also given that: the profit for each alpha pen is 8 Rs, and the profit for each beta pen is 5 Rs.

Therefore, the profit for x alpha pens is 8x, and the profit for y alpha pens is 5y.

Hence, the total profit 'p' = 8x + 5y

From equation (1), y = 500-2x and x = 250-0.5y.

Therefore, p = 8x + 5(500-2x) = 2500-2x      (2)

AND: p = 8(250-0.5y) + 5y = 2000+y             (3)

It is given that: the market can take only up to 150 alpha pens & 250 beta pens  per day from the company.

Therefore, $0\leq x \leq 150\\2200\leq p \leq 2500$

AND: $0\leq y\leq 250\\2000\leq p\leq 2250$.

Therefore, if we combine both conditions: $2200\leq p\leq 2250$

Therefore, max. profit is 2250 Rs, and min. profit is 2200 Rs.

Hence, substitute in eqn. (2): 2250 = 2500-2x

2x = 250

x = 125

Also, substitute in eqn. (3): 2250 = 2000+y

y = 250

Therefore, number of alpha pens produced = 125 and number of beta pens produced=250.