Chapter 1.7, Problem 49E

Solution

a.

To find: The function $y_1=u\left(x\right)$ that models the cost of producing $x$ unstrung rackets.

The function that models the cost of producing $x$ unstrung rackets is $y_1=23x+125000$ .

Given information:

The cost of making each unstrung racket is $23 and total of $125000 in fixed overhead costs.

Calculation:

Multiply the number of rackets with the price of each unstrung racket and add fixed overhead costs to get the function that models the cost of producing unstrung rackets as follows:

  $\begin{array}{c}{y}_1=\text{Cost of }x\text{ unstrung racket + fixed overhead}\\ =23\cdot x+125000\\ =23x+125000\end{array}$

Conclusion:

The function that models the cost of producing $x$ unstrung rackets is $y_1=23x+125000$ .

b.

To find: The function $y_2=s\left(x\right)$ that models the cost of producing $x$ strung rackets.

The function that models the cost of producing $x$ strung rackets is $y_2=31x+125000$ .

Given information:

The cost of making each unstrung racket is $23 and total of $125000 in fixed overhead costs.

Calculation:

Multiply the number of rackets with the price of each strung racket and add fixed overhead costs to get the function that models the cost of producing unstrung rackets as follows:

  $\begin{array}{c}{y}_2=\text{Cost of }x\text{ strung racket + fixed overhead}\\ =\left(23+8\right)\cdot x+125000\\ =31x+125000\end{array}$

Conclusion:

The function that models the cost of producing $x$ strung rackets is $y_2=31x+125000$ .

c.

To find: The function $y_3=R_u\left(x\right)$ that models the cost of producing $x$ unstrung rackets.

The function modelling the revenue generated by selling $x$ unstrung rackets is $y_3=56x$

Given information:

The cost of making each unstrung racket is $23 and total of $125000 in fixed overhead costs.

The price of an unstrung racket is $56 and the price of a strung racket is $79.

Calculation:

The selling price of one unstrung racket is $56. So, the selling price of $x$ unstrung rackets is $56x$ .

Therefore, the function modelling the revenue generated by selling $x$ unstrung rackets is $y_3=56x$ .

Conclusion:

The revenue function is $y_3=56x$ .

d.

To find: The function $y_4=R_s\left(x\right)$ that models the cost of producing $x$ strung rackets.

The function modelling the revenue generated by selling $x$ strung rackets is $y_4=79x$ .

Given information:

The cost of making each unstrung racket is $23 and total of $125000 in fixed overhead costs.

The price of an unstrung racket is $56 and the price of a strung racket is $79.

Calculation:

The selling price of one unstrung racket is $79. So, the selling price of $x$ strung rackets is $79x$ .

Therefore, the function modelling the revenue generated by selling $x$ unstrung rackets is $y_4=79x$ .

Conclusion:

The revenue function is $y_4=79x$ .

e.

To graph: The functions $y_1,y_2,y_3$ and $y_4$ on the same window.

Given information:

The cost of making each unstrung racket is $23 and total of $125000 in fixed overhead costs.

The price of an unstrung racket is $56 and the price of a strung racket is $79.

Graph:

Use a graphing tool to draw the graph of the functions as shown below.

  

Interpretation:

Selling strung rackets generates more revenue than selling unstrung rackets.

f.

To write: whether the company should manufacture unstrung or strung rackets.

The company should manufacture strung rackets.

Given information:

The cost of making each unstrung racket is $23 and total of $125000 in fixed overhead costs.

The price of an unstrung racket is $56 and the price of a strung racket is $79.

Calculation:

Consider the graph drawn in part (e).

  

From the graph it can be observed that the revenue generated by selling strung rackets is more than revenue generated by unstrung rackets. Thus, it is recommended to manufacture strung rackets.

Conclusion:

Selling strung rackets generates more revenue than selling unstrung rackets.