Jon 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

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

Answer 1

To find Jon's and Joey's elevations at different times, we will use their elevation expressions:

• Jon's elevation after t minutes: \( 225 - 10t \)
• Joey's elevation after t minutes: \( 50 + 15t \)

Now, let's calculate their elevations at 5, 10, and 15 minutes.

At 5 minutes:

• Jon's elevation: \[ 225 - 10(5) = 225 - 50 = 175 \text{ feet} \]
• Joey's elevation: \[ 50 + 15(5) = 50 + 75 = 125 \text{ feet} \]

At 10 minutes:

• Jon's elevation: \[ 225 - 10(10) = 225 - 100 = 125 \text{ feet} \]
• Joey's elevation: \[ 50 + 15(10) = 50 + 150 = 200 \text{ feet} \]

At 15 minutes:

• Jon's elevation: \[ 225 - 10(15) = 225 - 150 = 75 \text{ feet} \]
• Joey's elevation: \[ 50 + 15(15) = 50 + 225 = 275 \text{ feet} \]

Now we can put these values into the table.

| Minutes | Jon's elevation | Joey's elevation | |---------|------------------|------------------| | 0 | 225 | 50 | | 5 | 175 | 125 | | 10 | 125 | 200 | | 15 | 75 | 275 |

Finding when they are at the same elevation

To find when they are at the same elevation, we set their elevation expressions equal to each other:

\[ 225 - 10t = 50 + 15t \]

Now solve for \( t \):

1. Combine like terms: \[ 225 - 50 = 10t + 15t \] \[ 175 = 25t \]
2. Divide both sides by 25: \[ t = \frac{175}{25} = 7 \]

Thus, they will be at the same elevation after 7 minutes.

Checking the elevations at 7 minutes:

• Jon's elevation at 7 minutes: \[ 225 - 10(7) = 225 - 70 = 155 \text{ feet} \]
• Joey's elevation at 7 minutes: \[ 50 + 15(7) = 50 + 105 = 155 \text{ feet} \]

Both elevations are equal at 155 feet after 7 minutes.