Question

12. Ellen is training to compete in a half marathon. She currently runs at

a rate of 4.5 miles per hour.
a. Create a table and a graph showing Ellen's distance as it relates to
the amount of time she runs.
b. Write an equation to represent the relationship between the time
she runs and the distance she runs.
C. Is the equation from part b a direct variation equation? Explain
your reasoning.
d. If Ellen runs 13.1/miles to complete the half marathon, how long
will it take her to finish the race?

Answer 1

A.

`\begin{array}{cc}x&y\\ \\1&4.5\\2&9\\3&13.5\\4&18\end{array}`

B.
`y=4.5x`

C. See explanation

D. 2.911 hours

Explanation:

Ellen is training to compete in a half marathon. She currently runs at a rate of 4.5 miles per hour.

A. Let x be the time (in hours) and y be the distance (in miles). Then

`x=1\Rightarrow y=4.5\\ \\x=2\Rightarrow y=4.5\cdot 2=9\\ \\x=3\Rightarrow y=4.5\cdot 3=13.5\\ \\x=4\Rightarrow y=4.5\cdot 4=18\\ \\...`

Fill these values into the table

`\begin{array}{cc}x&y\\ \\1&4.5\\2&9\\3&13.5\\4&18\end{array}`

B. The equation that represents the relationship between the time she runs and the distance she runs is

`y=4.5x`

C. A direct variation describes a simple relationship between two variables . y varies directly with x if
`y=kx.`

Since the equation of the relationship is
`y=4.5x,` the equation from part B is a direct variation equation.

D. If Ellen runs 13.1 miles to complete the half marathon, then y = 13.1 miles. Find time x:

`13.1=4.5x\\ \\x=(13.1)/(4.5)=(131)/(45)\approx 2.91 \ hours`