Chapter 6, Problem 10MPS

To determine

To Solve:

The given problem.

Answer to Problem 10MPS

The minimum amount of sales to avoid any loss is $10,000.

Explanation of Solution

Given:

Given statement is :

A company’s monthly sales of $50,000 result in $6000 profit, while monthly sales of $70,000 result in $9000 profit.If profit is considered as a linear function of sales , what minimum amount of sales is needed to avoid a loss?

Calculation:

Let

p= profit

s = sales

Given that profit is considered as a linear function of sales .

So, $p=as+b\mathrm{..........}(1)$ where a and b are slope and p-intercept respectively.

We have p = 6000 and s = 50,000

  $\begin{array}{l}So,\\ 6000=50,000a+b\mathrm{.................}(2)\end{array}$

We have p = 9000 and s = 70,000

  $\begin{array}{l}So,\\ 9000=70,000a+b\mathrm{.................}(3)\end{array}$

Solving equation (2) and (3)

  $\begin{array}{l}50,000a+b=\mathrm{6000.........}(2)\\ 70,000a+b=\mathrm{9000.........}(3)\\ Subtract\text{ (2) from (3)}\\ \text{20,000}a=3000\\ \Rightarrow a=\frac{3000}{20,000}\mathrm{....................}[\text{divide each side by 20,000]}\\ \Rightarrow a=\frac{3}{20}\\ Substitute\text{ in equation (2)}\\ 50,000\left(\frac{3}{20}\right)+b=6000\\ \Rightarrow 7500+b=6000\\ \Rightarrow b=6000-\mathrm{7500...............}[\text{subtract 7500 from each side]}\\ \Rightarrow b=-1500\end{array}$

Substitute the values of a and b in equation (1):

  $p=\frac{3}{20}s-1500$

To avoid a loss, we need the profit to be non-negative , so minimum amount of sales to avoid a loss can be found out by substituting p = 0 , that is , when there is neither profit nor loss:

  $\begin{array}{l}p=\frac{3}{20}s-1500\\ p=0:\\ \Rightarrow 0=\frac{3}{20}s-1500\\ \Rightarrow 1500=\frac{3}{20}s\mathrm{..................}[\text{add 1500 each side]}\\ \Rightarrow 3s=1500(20)\mathrm{..............}[\text{multiply 20 each side]}\\ \Rightarrow 3s=30,000\\ \Rightarrow s=\frac{30,000}{3}\mathrm{.................}[\text{divide each side by 3]}\\ \Rightarrow s=10,000\end{array}$

So, the minimum amount of sales to avoid any loss is $10,000.