You and your friend part at an intersection. You drive off north at a constant speed, and your friend drives east at a speed that is 10 mph higher. After 4 hours the distance between you and your friend is 318.04 miles. You have been driving at ? mph. (Round to the nearest mile).

This problem can be solved using the Pythagorean theorem, as the path of you and your friend forms a right triangle where:

- One leg represents the distance you drove north.
- The other leg represents the distance your friend drove east.
- The hypotenuse represents the distance between the two of you after 4 hours.

Let's define:
- \( x \) as your speed in miles per hour.
- Your friend's speed is \( x + 10 \) miles per hour.

After 4 hours:
- The distance you traveled is \( 4x \).
- The distance your friend traveled is \( 4(x + 10) \).

From the Pythagorean theorem, the total distance between you and your friend is:

\[
\text{Distance}^2 = (\text{Distance you traveled})^2 + (\text{Distance friend traveled})^2
\]

This gives us the equation:

\[
318.04^2 = (4x)^2 + [4(x + 10)]^2
\]

Now we solve this equation to find \( x \).

\[
318.04^2 = 16x^2 + 16(x + 10)^2
\]
\[
101,112.9616 = 16x^2 + 16(x^2 + 20x + 100)
\]
\[
101,112.9616 = 16x^2 + 16x^2 + 320x + 1600
\]
\[
101,112.9616 = 32x^2 + 320x + 1600
\]
\[
99,512.9616 = 32x^2 + 320x
\]

Now, let's solve this quadratic equation.The solutions to the equation are approximately \( x = -60.99 \) and \( x = 50.99 \). Since speed cannot be negative, the correct speed at which you were driving is approximately \( 51 \) mph.

Would you like more details or have any further questions?

Here are 5 related questions for practice:
1. How would the solution change if your friend's speed was 15 mph faster instead of 10 mph faster?
2. What is the distance your friend traveled after 4 hours?
3. How much farther did your friend travel than you did in this scenario?
4. What would the distance between you and your friend be after 5 hours if the speeds remained constant?
5. What would happen to the distance between you and your friend if you increased your speed by 5 mph?

**Tip:** When dealing with motion problems, the Pythagorean theorem is useful for finding distances when two objects move at right angles to each other.

Learn how to solve a problem involving motion where two drivers travel at different speeds in perpendicular directions. This detailed step-by-step solution applies the Pythagorean Theorem and Quadratic Equation to calculate the correct speed. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation