Question

Amalia and Cindy leave from the same point at the same time. Cindy bicycles east at a rate that is 2 miles per hour faster than Amalia, who bicycles south. After one hour they are 10 miles apart. a. Let \(r\) represent Amalia's rate in miles per hour. Write an expression for the distance each girl has traveled in one hour. b. Draw and label a diagram of the situation. c. Use the Pythagorean theorem to find how fast each person is traveling. d. Which method did you use to solve the quadratic equation? Give a reason for your choice.

Short answer

Amalia is riding her bicycle at a speed of 6 mph and Cindy is riding her bicycle at a speed of 8 mph.

Step-by-step solution

Express distance in terms of rate

By definition, distance is equal to speed times time. In this case, Amalia's distance is \(r \times 1\) and Cindy's distance is \((r + 2) \times 1\).

Draw a diagram

Create a right-angled triangle with the starting point as the origin. Let the right-angle edge represent the path of Amalia and the horizontal edge represent the path of Cindy. The hypotenuse, they are 10 miles apart after 1 hour.

Use the Pythagorean theorem

Recall that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). That is, \(a^2 + b^2 = c^2\). Here, \(a=r\) (Amalia's distance), \(b=r+2\) (Cindy's distance), and \(c=10\) miles (their separation). Therefore, we have the equation \(r^2 + (r+2)^2 = 10^2\). This is a quadratic equation in terms of \(r\).

Solve the quadratic equation

Solving the equation \(r^2 + (r+2)^2 = 100\) gives a value of \(r = 6\) and a value of \(r = -8\). However, as the rate cannot be negative, the solution is \(r = 6\). Thus, Amalia is moving at a speed of 6 mph and Cindy is moving at 8 mph.

Discuss the method of solving the quadratic equation

Used the factoring method to solve the quadratic equation because there are no square roots in the coefficients and factors of the quadratic equation can be easily identified.