```Question 614383
Let's define:
.
For part (a) the annual linear total cost function as C(x)
For part (b) the annual linear total revenue function as R(x) and
For part (c) the annual linear profit function as P(x)
.
In part a, the cost of production will increase by \$23 for each unit that is produced. So to find the cost for the units produced, you just multiply the cost per unit times the number of units that are made. That is \$23 times x or just 23x. But the total cost also includes the annual fixed costs of \$400,000 probably consisting of taxes on the property, average annual building maintenance costs, average annual utility costs, and other similar costs that are incurred whether or not you produce any products. The total annual costs are the sum of these two. So we write the annual cost function in dollars as follows:
.
C(x) = 23x + 400,000
.
That's the answer to part (a)
.
For part (b) the total annual revenue function is the entire amount of money that the company receives for selling units of the product during the year. Since the selling price is is \$55 per unit, and the company sells x units during the year, the total revenue for the year is \$55 times x. We can express this as follows:
.
R(x) = 55x
.
and that's the answer to part (b)
.
But all the revenue is not profit. The profit is found by subtracting from the income (total revenue) the cost of production. In other words the annual profits is calculated from:
.
P(x) = R(x) - C(x)
.
But we now know what R(x) and C(x) are and we can substitute those relationships into this equation as follows:
.
P(x) = 55x - (23x + 400,000)
.
Since the parentheses are preceded by a minus sign, we can remove them if we change the signs of each of the terms within the parentheses. This makes the equation become:
.
P(x) = 55x - 23x - 400,000
.
We can then combine the 55x and -23x to get 32x and the equation is then:
.
P(x) = 32x - 400,000
.
And that's the answer to part (c)
.
Just as an interesting exercise, we weren't asked but we might want to find how many units the company has to sell each year, just to break even. By breaking even we mean that the company neither loses nor makes any money for the year. At this point the profit would be zero. So we can go to the profit equation and set P(x) equal to zero. The equation then becomes:
.
0 = 32x - 400,000
.
Subtract 32x from both sides and you have:
.
-32x = -400,000
.
Solve for x by dividing both sides by -32 and you get:
.
x = -400,000/-32 = 12,500
.
This means that unless the company sells 12,500 units in a year, it will lose money. Selling more than 12,500 units will make the company profitable.

Hope this helps you to understand the problem and how it can be worked.

```