if Vee walks the Grand Canyon at a rate of 1.5 mph uphill and 2 mph downhill, and it takes 5 hours longer to go up then come down, how far was the summit from the starting point?

To solve this problem, let’s define the following variables and equations:

- Let \( d \) be the distance (in miles) from the starting point to the summit.
- Let \( t \) be the time (in hours) it takes Vee to descend the distance.

### Step 1: Write Equations for Time
Vee walks:
- *Up* the Grand Canyon at 1.5 mph, so the time taken for the uphill trip is \( \frac{d}{1.5} \).
- *Down* the Grand Canyon at 2 mph, so the time taken for the downhill trip is \( \frac{d}{2} \).

### Step 2: Relate the Times
We know that it takes 5 hours longer for Vee to go up than to come down. Thus, we can write:

\[
\frac{d}{1.5} = \frac{d}{2} + 5
\]

### Step 3: Solve for \( d \)
To eliminate the fractions, find the least common multiple of 1.5 and 2, which is 6. Multiply both sides of the equation by 6:

\[
6 \cdot \frac{d}{1.5} = 6 \cdot \left(\frac{d}{2} + 5\right)
\]

This simplifies to:

\[
4d = 3d + 30
\]

### Step 4: Simplify and Solve
Subtract \( 3d \) from both sides:

\[
d = 30
\]

So, the distance from the starting point to the summit is **30 miles**.

### Final Answer:
**Vee was 30 miles away from the starting point at the summit.**

Would you like more details or have any questions?

Here are 5 related questions to deepen understanding:
1. How would the answer change if Vee walked downhill at 1 mph instead?
2. What would be the new distance if it took Vee only 3 hours longer to go uphill?
3. How can you set up a similar problem if Vee takes 4 hours longer uphill, but the rates are reversed?
4. How does the problem change if Vee's uphill speed was 2.5 mph and downhill speed was 3 mph?
5. How would you graph the relationship between time and distance for Vee’s journey?

**Tip:** When dealing with problems that involve different speeds and time differences, set up equations based on distance-time relationships and then solve systematically.

If Vee walks the Grand Canyon at a rate of 1.5 mph uphill and 2 mph downhill, and it takes 5 hours longer to go up than come down, how far was the summit from the starting point?

Discover how to solve a distance-time problem involving Vee's hike at different speeds up and down the Grand Canyon. Learn to set up equations based on rates, distances, and time differences. Suitable for students in Grades 7-10, this problem reinforces algebraic concepts and practical applications of speed and time calculations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation