Chapter 8.5, Problem 1CYU

To determine

To find: the constant rate of change and determine whether a proportional linear relationship exists between the two quantities.

Answer to Problem 1CYU

Rate of change is $\frac{1}{2}$ in. /week.

The linear relationship is not proportional.

Explanation of Solution

Given:

  

Consider three points from the graph that shows relationship between number of weeks and height of plants in inches: (0,3),(2,4) and (6,6)

Calculation:

In order to find constant rate of change, take any two points and use the formula for rate of change.

  $Rateofchange=\frac{changeiny}{changeinx}$

It means rate of change is change in height of plants with respect to change in number of weeks.

So, consider two points as: (0,3),(2,4)

Thus, constant rate of change is:

  $\begin{array}{l}Rateofchange=\frac{4-3}{2-0}\\ =\frac{1}{2}\end{array}$

Hence, rate of change is $\frac{1}{2}$ in. /week. It means that plant is growing at the rate of $\frac{1}{2}$ inches per week.

Linear relationships are called proportional if the ration of each non zero y-value compared to corresponding x-value is the same.

Now, it means that calculate ratio $\frac{height}{time}$ for each paired values.

For the point (0,3) the ratio is: $\frac{3}{0}=$ not defined.

And, for the point (2,4) the ratio is: $\frac{4}{2}=2$

And also, for the point (6,6) the ratio is: $\frac{6}{6}=1$

Hence, ratio $\frac{height}{time}$ is not same for each paired value.

Thus, the linear relationship is not proportional

Conclusion:

Therefore, rate of change is $\frac{1}{2}$ in. /week.

The linear relationship is not proportional.