Chapter 3.1, Problem 47AYU

Cost Function The simplest cost function is the linear cost function, $C(x)\text{ = }mx+b$, where the $y\text{-intercept}$ $b$ represents the fixed costs of operating a business and the slope $m$ represents the cost of each item produced. Suppose that a small bicycle manufacturer has daily fixed costs of $\$90$, and each bicycle costs $\$2000$ to manufacture.

a. Write a linear model that expresses the cost $C$ of manufacturing $x$ bicycles in a day.

b. Graph the model.

c. What is the cost of manufacturing 14 bicycles in a day?

d. How many bicycles could be manufactured for $\$3780$?

To determine

To calculate:

a. Write a linear model that expresses the cost $C$ of manufacturing $x$ bicycles in a day.

b. Graph the model.

c. What is the cost of manufacturing 14 bicycles in a day?

d. How many bicycles can be manufactured for $\$3780$?

Answer to Problem 47AYU

Solution:

a. $C(x)=90x+1800$

b. The cost of manufacturing 14 bicycles in a day is $\$3060$.

c. The graph is as shown below:

d. 22 bicycles can be manufactured from $\$3780$.

Explanation of Solution

Given:

The bicycle manufacturer has daily fixed costs of $\$1800$, and each bicycle costs $\$90$ to manufacture.

Formula used:

A linear equation is of the form $f(x)=mx+c$, where $m$ is the average rate if change of that function and $c$ is the $y\text{-intercept}$, which is the initial value of the function when $x=0$.

a.

The linear model that expresses the cost $C$ of manufacturing $x$ bicycles in a day is $C(x)=90x+1800$.

b.

The graph of the linear function is

c.

When $x=14$, we get

$C(14)=90(14)+1800=3060$

The cost of manufacturing 14 bicycles in a day is $\$3060$.

d.

When $C(x)=3780$, we get

$3780=90(x)+1800$

$\Rightarrow x=\frac{3780-1800}{90}=22$

Therefore, 22 bicycles can be manufactured from $\$3780$.