Chapter 3.1, Problem 47AYU

(a)

To determine

To write: a linear model that expresses the cost C of manufacturing x bicycles in a day.

Answer to Problem 47AYU

Therefore, the linear function that expresses the cost of manufacturing of x bicycles is $C\left(x\right)=90x+1800$ .

Explanation of Solution

Given:

  $C\left(x\right)=mx+b$

Calculation:

The linear cost function is given by $C\left(x\right)=mx+b$ , where b represents the fixed cost of operating business and m represents the cost of each item.

Now, the bicycle manufacturer has daily fixed costs of $1800. So bis 1800.

As the cost of each bicycle is $90, m is 90.

Substitute 1800 for b and 90 form in the equation $C\left(x\right)=mx+b$ .

  $C\left(x\right)=90x+1800$ .

Conclusion:

Therefore, the linear function that expresses the cost of manufacturing of x bicycles is $C\left(x\right)=90x+1800$ .

(b)

To determine

To graph: the model

Answer to Problem 47AYU

The graph was drawn.

Explanation of Solution

Calculation:

Compare the equation $C\left(x\right)=90x+1800$ with the standard linear equation $y=mx+b$ to find the slope and the y -intercept. Thus, 90 is the slope and 1800 is the y -intercept.

To plot the graph, take the y -intercept point $\left(0,1800\right)$ as one point. Since the slope of the line is 90, move 1 unit horizontally to the right and 90 units vertically upwards to find an additional point. So, the other point is $\left(1,1890\right)$ .

The graph can be plotted as follows.

  

Conclusion:

Therefore, thegraph was drawn.

(c)

To determine

To find: the cost of manufacturing 14 bicycles in a day.

Answer to Problem 47AYU

The cost of manufacturing of 14 bicycles is $3060.

Explanation of Solution

Calculation:

Substitute 14 for x in $C\left(x\right)=90x+1800$ .

  $\begin{array}{l}C\left(14\right)=90\cdot 14+1800\hfill \\ \begin{array}{l}=1260+1800\\ =3060\end{array}\hfill \end{array}$

Conclusion:

Therefore, the cost of manufacturing of 14 bicycles is $3060.

(d)

To determine

To find: number of bicycles could be manufactured for $3780.

Answer to Problem 47AYU

22 bicycles can be manufactured for $3780.

Explanation of Solution

Calculation:

Substitute 3780 for $C\left(x\right)$ in $C\left(x\right)=90x+1800$ .

  $3780=90x+1800$

Subtract 1800 from both sides of the equation $3780=90x+1800$ .

  $\begin{array}{l}3780-1800=90x+1800-1800\hfill \\ 1980=9x\hfill \end{array}$

Swap the sides of the equation.

  $9x=1980$

Divide both the sides by 9.

  $\begin{array}{l}\frac{9x}{90}=\frac{1980}{90}\\ x=22\end{array}$

Conclusion:

Therefore, 22 bicycles can be manufactured for $3780.