Christian and Jillian are hiking. Christian starts at an elevation of 375 feet, and is hiking down a mountain at a rate of 5 feet per minute, so his elevation is decreasing 5 feet every minute. At the same time, Jillian starts at an elevation of 25 feet and is hiking up at a rate of 2 feet per minute, so her elevation is increasing at a rate of 2 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.
Christian's elevation after t minutes:
375+(−5t)
Jillian's elevation after t minutes:
25+2t
Minutes Christian's elevation Jillian's elevation
0 375 25
20
40
60

Answer 1

We can use the expressions for both hikers to find their elevations at the specified times.

1. Christian's elevation after \( t \) minutes: \[ \text{Christian's elevation} = 375 - 5t \]

2. Jillian's elevation after \( t \) minutes: \[ \text{Jillian's elevation} = 25 + 2t \]

Now, let's compute their elevations for each time given.

For \( t = 20 \) minutes:

• Christian's elevation: \[ 375 - 5 \times 20 = 375 - 100 = 275 \]
• Jillian's elevation: \[ 25 + 2 \times 20 = 25 + 40 = 65 \]

For \( t = 40 \) minutes:

• Christian's elevation: \[ 375 - 5 \times 40 = 375 - 200 = 175 \]
• Jillian's elevation: \[ 25 + 2 \times 40 = 25 + 80 = 105 \]

For \( t = 60 \) minutes:

• Christian's elevation: \[ 375 - 5 \times 60 = 375 - 300 = 75 \]
• Jillian's elevation: \[ 25 + 2 \times 60 = 25 + 120 = 145 \]

Now, let's summarize the results in the table:

| Minutes | Christian's elevation | Jillian's elevation | |---------|-----------------------|---------------------| | 0 | 375 | 25 | | 20 | 275 | 65 | | 40 | 175 | 105 | | 60 | 75 | 145 |

Finding When They Are at the Same Elevation

To find out when they will be at the same elevation, we set the two expressions equal to each other:

\[ 375 - 5t = 25 + 2t \]

Now, we'll solve for \( t \):

1. Combine like terms: \[ 375 - 25 = 5t + 2t \] \[ 350 = 7t \]

2. Solve for \( t \): \[ t = \frac{350}{7} = 50 \text{ minutes} \]

Now, let’s check their elevations at \( t = 50 \) minutes:

• Christian's elevation: \[ 375 - 5 \times 50 = 375 - 250 = 125 \]
• Jillian's elevation: \[ 25 + 2 \times 50 = 25 + 100 = 125 \]

Both hikers are at the same elevation after \( 50 \) minutes, which is \( 125 \) feet.