A rubber ball is dropped from an unknown height. It bounces back to 60% of its original height. Write an equation you can use to find height.
To find the height from which the rubber ball is dropped, we can use the equation for the height of a bouncing ball.
Let's assume that "h" represents the height from which the rubber ball is dropped, and "b" represents the bouncing ratio (in decimal form) which is given as 60% or 0.6 of the original height.
When the ball bounces, it reaches a height of bh. On the subsequent bounce, it reaches a height of b^2h, and so on.
The equation to find the height can be expressed as:
h + bh + b^2h + b^3h + ... = h(1 + b + b^2 + b^3 + ...)
This formula represents an infinite geometric series since the ratio of each term to the previous term is constant and less than 1.
As per the formula for the sum of an infinite geometric series, the sum S can be calculated using the equation:
S = a / (1 - r)
Where "a" is the first term and "r" is the common ratio.
In this case, "a" is the height from which the ball is dropped (h), and "r" is the bouncing ratio (b). Thus, the equation becomes:
S = h / (1 - b)
Since the sum of an infinite geometric series represents the total distance traveled by the ball, we can use this equation to find the height (h) from which the ball is dropped.