Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Two kids at a summer camp, Jonah and Mabel, are competing in a potato sack race. Jonah is younger, so he is given a head start of $30$ meters. When the race starts, Jonah hops at a rate of $2$ meters per second, and Mabel hops $3$ meters per second. Eventually, Mabel will overtake Jonah. How long will that take? How far will Mabel have to hop? $\newline$ It will take $\_$ seconds for Mabel to hop $\_$ meters and catch up to Jonah.

Full solution

Q. Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.

\newline

Two kids at a summer camp, Jonah and Mabel, are competing in a potato sack race. Jonah is younger, so he is given a head start of

30

meters. When the race starts, Jonah hops at a rate of

2

meters per second, and Mabel hops

3

meters per second. Eventually, Mabel will overtake Jonah. How long will that take? How far will Mabel have to hop?

\newline

It will take

\_

seconds for Mabel to hop

\_

meters and catch up to Jonah.

Define Equations: Let's define two equations to represent the distances that Jonah and Mabel travel over time. Let $t$ be the time in seconds after the race starts. $\newline$ For Jonah, who starts $30$ meters ahead and hops at a rate of $2$ meters per second, the distance he travels can be represented by: $\newline$ $D_j = 2t + 30$
Calculate Distances: For Mabel, who starts from the starting line and hops at a rate of $3$ meters per second, the distance she travels can be represented by: $\newline$ $D_m = 3t$
Set Equations Equal: We want to find the time $t$ when Mabel catches up to Jonah. This happens when their distances are equal, so we set the two equations equal to each other: $\newline$ $2t + 30 = 3t$
Solve for Time: Now we solve for $t$ by subtracting $2t$ from both sides of the equation: $\newline$ $2t + 30 - 2t = 3t - 2t$ $\newline$ $30 = t$
Calculate Mabel's Distance: We have found that it will take $30$ seconds for Mabel to catch up to Jonah. Now we need to find out how far Mabel will have hopped in that time. We use Mabel's distance equation with $t = 30$ : $\newline$ $D_m = 3t$ $\newline$ $D_m = 3(30)$ $\newline$ $D_m = 90 \, \text{meters}$