Chapter 3.1, Problem 48AYU

Cost Function Refer to Problem 47. Suppose that the landlord of the building increases the bicycle manufacturer's rent by $\$100$ per month.

a. Assuming that the manufacturer is open for business 20 days per month, what are the new daily fixed costs?

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

c. Graph the model.

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

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

To determine

To calculate:

a. Assuming that the manufacture is open for business 20 days per month, what are the new daily fixed costs?

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

c. Graph the model.

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

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

Answer to Problem 48AYU

Solution:

a. $\$1805$

b. $C(x)=90x+1805$

c. The graph is as shown below:

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

e. 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.

The building rent is increased by $\$100$ per month and the manufacturer works for 20 days in a month.

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 $x=0$.

Calculation:

a.

Since, the building rent is increased by $\$100$ for 20 days, the building rent increased for one day will be $\frac{100}{20}=5$ dollars per day.

Therefore, the new daily fixed cost for the manufacturing is $\$1800+\$5=\$1805$.

b.

The new linear model that expresses the cost $C$ of manufacturing $x$ bicycles in a day is $C(x)=90x+1805$.

c.

The graph of the linear function is

d.

When $x=14$, we get

$C(14)=90(14)+1805=3065$

The cost of manufacturing 14 bicycles in a day is $\$3065$.

e.

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

$3780=90(x)+1805$

$\Rightarrow x=\frac{3780-1805}{90}=21.9\approx 22$

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