Question

A shoe manufacturer is planning production of a new variety of shoes. For the one year the fixed cost for setting up the new production line is ₨1.25 lakh. Variable cost for producing each pair of shoes is ₨35. The sales department projects that 1500 pairs can be sold in the first year at the rate of ₨160 pairs.
1) Determine the cost function \[{\text{C(x)}}\] for the total cost of producing x pairs of shoes.
2) Determine the revenue function \[{\text{R(x)}}\] for the total revenue from the sales of x pairs of shoes.
3) Determine the profit function \[{\text{P(x)}}\]for the profit from the sale of x pairs of shoes.
4) If 1500 pairs are actually sold what profit or loss the company would incur?

Answer 1

Hint:
In this question, we have to determine the cost price, revenue generated, and profit function. Here we will use the relation of total cost with cost function, revenue function and profit function to find the answer.

Complete step by step solution:
1) As we all know that the total cost of producing a product is equal to its fixed plus the variable cost associated with it.
So, cost function is
\[{\text{C(x)}} = 125000 + 35{\text{x}}\]
where, x is the number of shoes produced
2) So, as we all know that the total revenue of a product is equal to the total amount generated by selling the product at some selling price
So, revenue function is
\[{\text{R(x)}} = 160{\text{x}}\]
where, x is the number of shoes sold
3) So, as we all know that the total profit generated by selling a product is equal to the difference between the revenue generated and the total cost of production
So, profit function is
\[{\text{P(x)}} = {\text{R(x)}} - {\text{C(x)}} = 160{\text{x}}-\left( {125000 + 35{\text{x}}} \right)\]
Simplifying the expression, we get
\[{\text{P(x)}} = 125{\text{x}}-125000\]
where, x is the number of shoes sold
4) We will now find the profit or loss of a company we need a profit function which we already derived in the above equation.
So, by putting the value of number of shoes sold in the above equation, \[{\text{x}} = {\text{1500}}\], we get
\[{\text{P(x)}} = 125 \times (1500)-125000 = 62500\]

So, the profit earned by the company is ₨62500

Note:
Cost price is the cost of producing something or the price at which it is sold without making any money. Selling price is the price at which something is sold. Revenue of a product is equal to the total amount generated by selling the product at some selling price. Profit is the money that you make when you sell something for more than it cost you. If the revenue generated by the company is less than the total cost of production then there is a loss to the company rather than profit.