Chapter 1.2, Problem 16E

(a)

To determine

To express: The cost in terms of the number of produced chairs and sketch the graph of the cost function.

Answer to Problem 16E

The equation of the temperature in terms of the number of chirps per minute N is $y=13x+900$.

Explanation of Solution

Let x axis be represented the number of chairs per day and y axis be represented the manufacturing cost in dollars.

Recall the general equation of the linear function $y=mx+c$ where m is the slope and c is the y-intercept.

Since the cost function follows the linear function, the equation of the cost y in terms of the number of produced chairs x is also in the form of $y=mx+c$.

According to the given data, there are two points such as (100, 2200) and (300, 4800).

Obtain the slope m by using the two points.

$\begin{array}{c}m=\frac{4800-2200}{300-100}\\ =\frac{2600}{200}\\ =13\end{array}$

Thus, the slope $m=13$.

Use the slope $m=13$, any one the two point say (100, 2200) and obtain the equation of cost y in terms of x as follows.

$\begin{array}{l}y-y_1=m\left(x-x_1\right)\left[\text{Slope-point}\text{form}\right]\\ y-2200=13\left(x-100\right)\\ y=13x-1300+2200\\ y=13x+900\end{array}$

Thus, the required equation is $y=13x+900$.

The graph of cost function is shown below in Figure 1.

From Figure 1, it is observed that the graph is a straight line as the function is linear.

(b)

To determine

To find: The slope of the graph of the cost function and interpret it.

Answer to Problem 16E

The slope of the graph of the cost function is $m=13$. It represents the rate of change of cost with respect to the number of chairs produced.

Explanation of Solution

From part (a), the equation of cost as the function of number of chairs produced per day is $y=13x+900$.

Since it follows a linear function, the coefficient of x is considered as slope.

Thus, the required slope is $m=13$.

Also, notice that the cost increases as the number of chairs produced per day increases as it follows the direct variation.

Observe that, if the number of produced chairs increases by 200, then the cost increases $2600. That is 13 chairs per dollar.

Therefore, the slope represents the rate of change of cost with respect to the number of chairs produced per day.

(c)

To determine

To find: The y-intercept of the graph of the cost function; interpret what it represent.

Answer to Problem 16E

The y-intercept of the graph of the cost function is 900. It represents the fixed manufacturing cost per day.

Explanation of Solution

The equation of cost as the function of number of chairs per day is $y=13x+900$.

Since it follows a linear function, the constant term c is considered as y-intercept.

Thus, the y-intercept is $900$.

The y-intercept represents the fixed manufacturing cost per day.