on and Joey are hiking. Jon starts at an elevation of 225 feet, and is hiking down a mountain at a rate of 10 feet per minute, so his elevation is decreasing 10 feet every minute. At the same time, Joey starts at an elevation of 50 feet and is hiking up at a rate of 15 feet per minute, so his elevation is increasing at a rate of 15 feet every minute. The variable t represents the time in minutes they have been hiking. When will the two hikers be at the same elevation?

Question

on and Joey are hiking. Jon starts at an elevation of 225 feet, and is hiking down a mountain at a rate of 10 feet per minute, so his elevation is decreasing 10 feet every minute. At the same time, Joey starts at an elevation of 50 feet and is hiking up at a rate of 15 feet per minute, so his elevation is increasing at a rate of 15 feet every minute. The variable t represents the time in minutes they have been hiking. When will the two hikers be at the same elevation?
We can use a table below to record the hikers’ elevations after given times. The first row has been completed for you: after 0 minutes, the hikers are at their starting elevations. Use the expressions for each hiker to find their elevations after each of these times. You can use the calculator to help you solve.
Jon's elevation after t minutes:
225+(−10t)
Joey's elevation after t minutes:
50+15t
Minutes	Jon's elevation	Joey's elevation
0	225	50
5	
10	
15	
Nice!
Use the table to estimate. About when will the two hikers be at the same elevation? In other words, after about how many minutes will Jon be at the same elevation as Joey?
Jon's elevation after t minutes: 225+(−10t)
Joey's elevation after t minutes: 50+15t
Minutes	Jon's elevation	Joey's elevation	
0	225	50	
5	175	125	
10	125	200	
15	75	275

Between 5 and 10

Between 10 and 15

Between 0 and 5
Great job!
Now that we’ve estimated when the hikers will be at the same elevation, let’s find the exact answer. You can set the expressions for Jon’s elevation and Joey’s elevation equal to one another to write an equation that represents the two hikers being at an equal elevation. Do that now.
Jon's elevation after t minutes: 225+(−10t)
Joey's elevation after t minutes: 50+15t

=

Jon's elevation
Joey's elevation

GPT-4o mini · Answer

To find when Jon and Joey will be at the same elevation, we can set their elevation expressions equal to one another.

Jon's elevation after \( t \) minutes is given by: \[ 225 - 10t \]

Joey's elevation after \( t \) minutes is given by: \[ 50 + 15t \]

Setting these two expressions equal, we have: \[ 225 - 10t = 50 + 15t \]

Now, we can solve for \( t \).

First, rearranging the equation: \[ 225 - 50 = 10t + 15t \] \[ 175 = 25t \]
Now, divide both sides by 25 to isolate \( t \): \[ t = \frac{175}{25} = 7 \]

Therefore, Jon and Joey will be at the same elevation after \( t = 7 \) minutes.

Summary

Jon's and Joey's elevations will be equal after 7 minutes of hiking.