Chapter 2.2, Problem 27E

a.

To determine

To find: the equation representing cost function.

Answer to Problem 27E

  $C\left(x\right)=50x+2000$

Explanation of Solution

Given information:

Cost of producing 20 color television = $3000

Cost of producing 60 color television = $5000

Calculation:

Let the number of color television, their cost, and cost function be x, y and $C\left(x\right)$ respectively. The point coordinates are $\left(20,3000\right)$ and $\left(60,5000\right)$ respectively.

Put the points in the slope formula and evaluate the value of m as follows:

  $\begin{array}{c}m=\frac{y_2-y_1}{x_2-x_1}\\ =\frac{5000-3000}{60-20}\\ =\frac{2000}{40}\\ =50\end{array}$

Now, put the value of m and any one point coordinate to evaluate the equation for cost as follows:

  $\begin{array}{c}y-y_0=m\left(x-x_0\right)\\ y-5000=50\left(x-60\right)\left[\text{Substituting }\left(x_0,y_0\right)\text{as }\left(60,5000\right)\right]\\ y-5000=50x-3000\\ y=50x-3000+5000\\ y=50x+2000\\ C\left(x\right)=50x+2000\left[\text{Rewritting the equation in terms of cost function}\right]\end{array}$

Thus, the equation for cost function is $C\left(x\right)=50x+2000$ .

b.

To determine

To find: the fixed cost and variable cost per unit.

Answer to Problem 27E

The fixed cost is 2000 and variable cost per unit is 50.

Explanation of Solution

Given information:

Cost of producing 20 color television = $3000

Cost of producing 60 color television = $5000

The cost function evaluated is as follows:

  $C\left(x\right)=50x+2000$

Calculation:

Compare the given equation with the slope intercept formula as follows:

  $\begin{array}{c}y=mx+b\\ C\left(x\right)=50x+2000\end{array}$

  $\therefore m=50\text{ and }b=2000$

Since the value of intercept is fixed and the slope is the rate of a line which changes which change in variable. Hence, the fixed cost is 2000 and variable cost per unit is 50.

c.

To determine

To graph: the cost function.

Answer to Problem 27E

  

Explanation of Solution

Given information:

Cost of producing 20 color television = $3000

Cost of producing 60 color television = $5000

The cost function evaluated is as follows:

  $C\left(x\right)=50x+2000$

Calculation:

Let $y=C\left(x\right)$ . Since, the cost cannot be negative. So, make a table of values for $y=50x+2000$ by taking positive value of x as shown below.

$x$	0	2	6	10
$y$	2000	2100	2300	2500

Plot the points on a graph for $C\left(x\right)=50x+2000$ as shown below.