A rubber ball dropped on a hard surface takes a sequence of bounces, each one 4/5 as high as the preceding one. If this ball is dropped from a height of 10 feet, what is the total vertical distance it has traveled at the time when it hits the surface for its fifth bounce?
To find the total vertical distance the ball has traveled at the time of its fifth bounce, we can sum up the distances it travels during each bounce.
When dropped from a height of 10 feet, the ball bounces to a height of 10 * 4/5 = 8 feet after the first bounce.
After the second bounce, the ball reaches a height of 8 * 4/5 = 6.4 feet.
After the third bounce, the ball reaches a height of 6.4 * 4/5 = 5.12 feet.
After the fourth bounce, the ball reaches a height of 5.12 * 4/5 = 4.096 feet.
After the fifth bounce, the ball reaches a height of 4.096 * 4/5 = 3.2768 feet.
So, the total vertical distance the rubber ball has traveled at the time of its fifth bounce is 10 + 8 + 6.4 + 5.12 + 4.096 + 3.2768 = 36.8768 feet.