Chapter 8.7, Problem 29E

a.

To determine

To write:

A function for your total costs, $c(x)$ .

Answer to Problem 29E

The function for total costs would be $c(x)=10x+3000$ .

Explanation of Solution

Given:

You make and sell birdhouses. Your fixed cost for your tools and workspace are $3000. The cost of wood and other materials needed to make a birdhouse is $10. You sell each birdhouse for $50.

Calculation:

Let x represent the number of birdhouses make and sell.

We have been given that cost of wood and other materials needed to make a birdhouse is $10, so cost of materials for x birdhouses would be 10x .

The total cost of making x birdhouses would be cost of materials plus fixed costs (3000).

  $c(x)=10x+3000$

Therefore, thefunction for total costs would be $c(x)=10x+3000$ .

b.

To determine

To write:

A function for your total income, $i(x)$ .

Answer to Problem 29E

The function for total income would be $i(x)=50x$ .

Explanation of Solution

Given:

You make and sell birdhouses. Your fixed cost for your tools and workspace are $3000. The cost of wood and other materials needed to make a birdhouse is $10. You sell each birdhouse for $50.

Calculation:

Let x represent the number of birdhouses make and sell.

We have been given that yousell each birdhouse for $50, so income from sellingx birdhouses would be 50x .

Therefore, thefunction for income would be $i(x)=50x$ .

c.

To determine

To write:

A function for your profit, $p(x)$ .

Answer to Problem 29E

The function for profit would be $p(x)=40x-3000$ .

Explanation of Solution

Given:

You make and sell birdhouses. Your fixed cost for your tools and workspace are $3000. The cost of wood and other materials needed to make a birdhouse is $10. You sell each birdhouse for $50.

Calculation:

We know that profit is difference of income and total costs.

To write the profit function, we will subtract cost function from income function as:

  $\begin{array}{l}p(x)=i(x)-c(x)\\ p(x)=50x-(10x+3000)\\ p(x)=50x-10x-3000\\ p(x)=40x-3000\end{array}$

Therefore, thefunction for profit would be $p(x)=40x-3000$ .

d.

To determine

To find:

What is the profit when you make and sell 100 birdhouses?

Answer to Problem 29E

You will make a profit of $1000.

Explanation of Solution

Given:

You make and sell birdhouses. Your fixed cost for your tools and workspace are $3000. The cost of wood and other materials needed to make a birdhouse is $10. You sell each birdhouse for $50.

Calculation:

To find the profitwhen you make and sell 100 birdhouses, we will substitute $x=100$ in our function as:

  $\begin{array}{l}p(x)=40x-3000\\ p(100)=40(100)-3000\\ p(100)=4000-3000\\ p(100)=1000\end{array}$

Therefore, you will make a profit of $1000.

e.

To determine

To find:

How many birdhouses do you need to make and sell in order to break even?

Answer to Problem 29E

You need to make and sell 75 birdhouses in order to break even.

Explanation of Solution

Given:

You make and sell birdhouses. Your fixed cost for your tools and workspace are $3000. The cost of wood and other materials needed to make a birdhouse is $10. You sell each birdhouse for $50.

Calculation:

We know that breakeven is a point, where you make a profit of $0.

To find the break even, we will equate profit function with 0 and solve for xas :

  $\begin{array}{l}p(x)=40x-3000\\ 40x-3000=0\\ 40x-3000+3000=0+3000\\ 40x=3000\\ \frac{40x}{40}=\frac{3000}{40}\\ x=75\end{array}$

Therefore, you need to make and sell 75 birdhouses in order to break even.