Mary and Anna start at 1200H at the same place and
travel along straight roads that are perpendicular to each
other. Mary travels 10 kph faster than Anna. At the end of
45 minutes, they are 63.86 kilometers apart. Solve for the
respective rates of Mary and Anna.

Let's define the following variables:

- Let \( v_A \) represent Anna's speed (in km/h).
- Let \( v_M \) represent Mary's speed (in km/h).

We are given:

- Mary travels 10 kph faster than Anna:  
  \[
  v_M = v_A + 10
  \]
- They are 63.86 kilometers apart after 45 minutes (or 0.75 hours).
- The roads are perpendicular, so their paths form a right triangle. The legs of the triangle are the distances traveled by Mary and Anna, and the hypotenuse is the distance between them (63.86 km).

Now, after 45 minutes (0.75 hours):

- The distance Mary travels is:
  \[
  d_M = v_M \times 0.75
  \]
- The distance Anna travels is:
  \[
  d_A = v_A \times 0.75
  \]

Since the roads are perpendicular, the Pythagorean theorem applies:
\[
d_M^2 + d_A^2 = 63.86^2
\]

Substituting \( d_M \) and \( d_A \) with the expressions for their distances:
\[
(v_M \times 0.75)^2 + (v_A \times 0.75)^2 = 63.86^2
\]

Now substitute \( v_M = v_A + 10 \) into the equation:
\[
((v_A + 10) \times 0.75)^2 + (v_A \times 0.75)^2 = 63.86^2
\]

We can solve this quadratic equation to find \( v_A \), and then calculate \( v_M \).

Let's solve it step by step.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like to go over the steps manually, or do you have any questions about the process so far?

Here are five related questions to expand on this problem:
1. How would the solution change if Mary was 15 kph faster than Anna?
2. What if the time was 30 minutes instead of 45 minutes?
3. How does the Pythagorean theorem apply to problems involving right-angled paths?
4. Could you use a different method, like graphical methods, to solve this problem?
5. How would the problem change if the roads were not perpendicular?

**Tip:** In problems involving right triangles and motion, the Pythagorean theorem is extremely useful for determining distances between moving objects.

Mary and Anna start at 1200H at the same place and travel along straight roads that are perpendicular to each other. Mary travels 10 kph faster than Anna. At the end of 45 minutes, they are 63.86 kilometers apart. Solve for the respective rates of Mary and Anna.

This problem explores how to solve a motion problem where two people travel on perpendicular roads. Mary travels 10 kph faster than Anna, and after 45 minutes, they are 63.86 kilometers apart. Using the Pythagorean theorem and distance formulas, find the rates of both Mary and Anna. Suitable for students in Grades 9-11.

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