Break-Even Point: Manufacturing Company Analysis

[Victoriaa]

A manufacturing company is studying the feasibility of producing a new product. A new production line could manufacture up to 800 units per month at a cost of \$50 per unit. Fixed costs would be \$22,400 per month. Variable selling and shipping costs are estimated to be \$20 per unit. Market research indicates that a unit price of \$110 would be competitive.
a) What is the break-even point as a percent of capacity?
b) What would be the net income at 90% of capacity?
c) What would unit sales have to be to attain a net income of \$9000 per month?
d) In a serious recession sales might fall to 55% of capacity. What would be the
resulting net income?
e) What dollar amount of sales would result in a loss of \$2000 per month?

 

[MarkFL]

Hello and welcome to MHB, Victoriaa! :D

In order for us to best help you, we need to know what you have tried so we can see where exactly you are having trouble. This is why we ask people to show effort.

But, having said that, in order to answer the questions, we need to figure out some functions, namely revenue, cost, and profit all defined on a monthly basis. Let's let $x$ be the number of units manufactured and sold during a given month.

And so revenue $R$ would be the product of the number of units sold and the price per unit (which we are told is \$110):

\(\displaystyle R(x)=110x\)

The cost function we obtain from the given information as follows:

\(\displaystyle C(x)=50x+22400+20x=70x+22400=70(x+320)\)

Profit, or net income, is defined as revenue minus cost, hence:

\(\displaystyle P(x)=110x-(70x+22400)=40x-22400=40(x-560)\)

a) The break-even point is where profit is 0, so what you need to do here is set $P(x)=0$ and solve for $x$. Then take the value you find for $x$ and divide it bt the capacity of 800 units to find what portion of capacity this $x$ is and then multiply by 100 to convert to a percentage. You could simply divide $x$ by 8 to get the percentage in one step:

\(\displaystyle \frac{x}{800}\cdot100=\frac{x}{8}\)

What do you find?

 

[MarkFL]

a) Setting $P(x)=0$, we have:

\(\displaystyle 40(x-560)=0\implies x=560\)

And so the percentage of capacity at the break-even point is then:

\(\displaystyle \frac{560}{8}=70\)

So, we conclude that in order to break even, production must be at 70% of capacity.

b) What would be the net income at 90% of capacity?

90% of capacity is:

\(\displaystyle \frac{9}{10}\cdot800=720\)

And so the net income (profit) (in dollars) at this production level is:

\(\displaystyle P(720)=40(720-560)=3200(9-7)=6400\)

c) What would unit sales have to be to attain a net income of \$9000 per month?

\(\displaystyle P(x)=40x-22400=9000\)

\(\displaystyle 40x=31400\)

\(\displaystyle x=785\)

d) In a serious recession sales might fall to 55% of capacity. What would be the resulting net income?

\(\displaystyle \frac{11}{20}\cdot800=440\)

\(\displaystyle P(440)=40(440-560)=1600(11-14)=-4800\)

At a product level of 55%, their would be a net loss of \$4800.

e) What dollar amount of sales would result in a loss of \$2000 per month?

First we need to find the unit sales for a loss of \$2000:

\(\displaystyle P(x)=40x-22400=-2000\)

\(\displaystyle 40x=20400\)

\(\displaystyle x=510\)

And so the revenue at this level (in dollars) is:

\(\displaystyle R(510)=110(510)=56100\)