Chapter 2, Problem 1RE

a.

To determine

The distance between the points $\left(0,0\right),\left(4,2\right)$.

Answer to Problem 1RE

The distance between the points $\left(0,0\right),\left(4,2\right)$ is $\underset{\_}{2\sqrt{5}}$.

Explanation of Solution

The given points are $\left(0,0\right),\left(4,2\right)$.

Obtain the distance between the points $\left(0,0\right),\left(4,2\right)$ as follows.

Substitute $\left(x_1,y_1\right)=\left(0,0\right)\text{ and }\left(x_2,y_2\right)=\left(4,2\right)$ in $d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$.

$\begin{array}{c}d=\sqrt{{\left(4-0\right)}^2+{\left(2-0\right)}^2}\\ =\sqrt{16+4}\\ =\sqrt{20}\\ =\sqrt{4\cdot 5}\\ =2\sqrt{5}\end{array}$

Thus, the distance between the points $\left(0,0\right),\left(4,2\right)$ is $\underset{\_}{2\sqrt{5}}$.

b.

To determine

The midpoint of the line segment connecting the points $\left(0,0\right),\left(4,2\right)$.

Answer to Problem 1RE

The midpoint of the line segment connecting the points $\left(0,0\right),\left(4,2\right)$ is $\underset{\_}{\left(2,1\right)}$.

Explanation of Solution

The given points are $\left(0,0\right),\left(4,2\right)$.

Obtain the midpoint of the line segment connecting the points $\left(0,0\right),\left(4,2\right)$ as follows.

Substitute $\left(x_1,y_1\right)=\left(0,0\right)\text{ and }\left(x_2,y_2\right)=\left(4,2\right)$ in $\left(x,y\right)=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$.

$\begin{array}{c}\left(x,y\right)=\left(\frac{0+4}{2},\frac{0+2}{2}\right)\\ =\left(\frac{4}{2},\frac{2}{2}\right)\\ =\left(2,1\right)\end{array}$

Thus, the midpoint of the line segment connecting the points $\left(0,0\right),\left(4,2\right)$ is $\underset{\_}{\left(2,1\right)}$.

c.

To determine

The slope of the line containing the points $\left(0,0\right),\left(4,2\right)$.

Answer to Problem 1RE

The slope of the line containing the points $\left(0,0\right),\left(4,2\right)$ is $\underset{\_}{m=\frac{1}{2}}$.

Explanation of Solution

The given points are $\left(0,0\right),\left(4,2\right)$.

Obtain the slope of the line containing the points $\left(0,0\right),\left(4,2\right)$ as follows.

Substitute $\left(x_1,y_1\right)=\left(0,0\right)\text{ and }\left(x_2,y_2\right)=\left(4,2\right)$ in $m=\frac{y_2-y_1}{x_2-x_1}$.

$\begin{array}{c}m=\frac{2-0}{4-0}\\ =\frac{2}{4}\\ =\frac{1}{2}\end{array}$

Thus, the slope of the line containing the points $\left(0,0\right),\left(4,2\right)$ is $\underset{\_}{m=\frac{1}{2}}$.

d.

To determine

To interpret: The slope found in part (c).

Explanation of Solution

From part (c), the slope of the line containing the points $\left(0,0\right),\left(4,2\right)$ is $m=\frac{1}{2}$.

For every 2 units changes in x, y will change by 1 unit.

That is, if x increases by 2 units, then y will decreases by 1 unit.

Thus, the average rate of change of y with respect to x is $\frac{1}{2}$