Question

An ice- cream company sells one of its types of ice cream by weight. If the product is not sold on the day it is prepared, it can be sold at a loss of Rs.5 per kg. But there is an unlimited market for one day ice cream. On the other hand, the company makes a profit of Rs. 12 per kg on every kg of ice cream sold on that day it is prepared. Past daily orders from a distribution with F(X) = 0.002-0.0002X, 0 ≤ X ≤ 100. How many kg of ice cream should the company prepare every day?

Answer 1

To determine the quantity of ice cream the company should prepare every day, we need to find the point where marginal revenue equals marginal cost.

This can be done by solving the equation F(X) = profit - loss, and substituting the values of profit and loss. The optimal quantity is found to be 39,990 kg of ice cream.

Step-by-step explanation:

To determine how many kilograms of ice cream the company should prepare every day, we need to find the quantity of ice cream that maximizes profit. We can do this by finding the point where the marginal revenue (MR) equals the marginal cost (MC).

In this case, the marginal revenue is given by the distribution F(X), and the marginal cost is the difference between the profit made on the day of sale and the loss incurred if the ice cream is not sold that day.

To find the optimal quantity, we need to solve the equation MR = MC, which gives us F(X) = profit - loss. We can substitute the values of profit and loss to get the equation in terms of X. Solving for X will give us the quantity of ice cream that should be prepared every day.

Let's work through an example. Assume the profit on the day of sale is Rs. 12 per kg and the loss is Rs. 5 per kg. We have F(X) = 0.002 - 0.0002X. Setting F(X) equal to the profit minus the loss gives us:

0.002 - 0.0002X = 12 - 5

Simplifying, we get -0.0002X = 7.998. Dividing both sides by -0.0002 gives us X = -7.998/-0.0002 = 39,990 kg.

Therefore, the company should prepare 39,990 kg of ice cream every day to maximize profit.