Chapter 4, Problem 9PT

(a)

To determine

To find:The equation of the line for the given condition in slope intercept form.

Answer to Problem 9PT

The equation of the straight line is $y=\frac{13}{10}x+30$ .

Explanation of Solution

Given:

The equation passes through points $\left(100,160\right)$ and $\left(500,680\right)$

Calculation:

The standard form of line passing through two points is $y-y_1=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)$ .

Here $\frac{y_2-y_1}{x_2-x_1}$ is the slope of the straight line. $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ are the points from which the straight line passes.

Substitute $\left(100,160\right)$ for $\left(x_1,y_1\right)$ and $\left(500,680\right)$ for $\left(x_2,y_2\right)$ .

The equation of the straight line is

  $\begin{array}{l}y-y_1=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)\\ y-\left(160\right)=\frac{\left(680\right)-\left(160\right)}{\left(500\right)-\left(100\right)}\left(x-100\right)\\ y-160=\frac{13}{10}\left(x-100\right)\\ y=\frac{13}{10}\left(x\right)-130+160\\ y=\frac{13}{10}x+30\end{array}$

Therefore, the equation of the straight line is $y=\frac{13}{10}x+30$ .

(b)

To determine

To predict:The amount of time required to paint a room of size 750 square feet.

Answer to Problem 9PT

The amount of time required to paint a room of size 750 square feetis $1005\text{ minutes}$ .

Explanation of Solution

Given:

The equation passes through points $\left(100,160\right)$ and $\left(500,680\right)$

Calculation:

The standard form of line passing through two points is $y-y_1=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)$ .

Here $\frac{y_2-y_1}{x_2-x_1}$ is the slope of the straight line. $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ are the points from which the straight line passes.

Substitute $\left(100,160\right)$ for $\left(x_1,y_1\right)$ and $\left(500,680\right)$ for $\left(x_2,y_2\right)$ .

The equation of the straight line is

  $\begin{array}{l}y-y_1=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)\\ y-\left(160\right)=\frac{\left(680\right)-\left(160\right)}{\left(500\right)-\left(100\right)}\left(x-100\right)\\ y-160=\frac{13}{10}\left(x-100\right)\\ y=\frac{13}{10}\left(x\right)-130+160\\ y=\frac{13}{10}x+30\end{array}$

The equation of the straight line is $y=\frac{13}{10}x+30$ .

Calculate the time required for the room of size 750 square feet. Substitute 750 for x and calculate the time in minutes given by y .

  $\begin{array}{l}y=\frac{13}{10}x+30\\ y=\frac{13}{10}\times 750+30\\ y=1005\text{ minutes}\text{.}\end{array}$

Therefore, the time required is $1005\text{ minutes}$ .