Question

A rainbow's path follows the quadratic r(x)-1/52(x+15)(x-47), where x is the horizontal distance in miles, and r(x) is the height of the rainbow, in miles. What is the distance between the two places where the rainbow appears to hit the ground?

Answer 1

To find the distance between the two places where the rainbow appears to hit the ground, we need to find the roots of the quadratic r(x) = (x+15)(x-47)/52.

The roots of a quadratic equation are the values of x that make the quadratic equal to zero. So, we need to solve the equation:

r(x) = 0

Substituting the expression for r(x), we get:

(x+15)(x-47)/52 = 0

This equation is true if and only if one of the factors is zero. Therefore, we need to solve the equations:

x + 15 = 0 or x - 47 = 0

The solutions to these equations are:

x = -15 or x = 47

Since we are interested in the distance between the two places where the rainbow appears to hit the ground, we need to take the absolute value of the difference between these two solutions:

|47 - (-15)| = 62

Therefore, the distance between the two places where the rainbow appears to hit the ground is 62 miles.

Explanation: