A rubber ball is dropped onto a hard surface from a height of 9 feet, and it bounces up and down. At each bounce, it rises to 80% of the height from which it fell.
a. Find a formula for \(h(n)\), the height in inches reached by the ball on bounce \(n\).
\[ h(n) = \]
b. How high will the ball bounce on the 10th bounce?
c. How many bounces before the ball rises no higher than an inch?
Answer (3)
There are 12 inches in a foot, so 9ft = 108in. Also, 80% = 0.8. Therefore the formula is: h(n) = 108 * 0.8^n.
To find the bounce height after 10 bounces, substitute n=10 into the equation: h(n) = 108 * 0.8^10 = 11.60in (2.d.p.).
Finally to find how many bounces happen before the height is less than one inch, substitute h(n) = 1, then rearrage with logarithms to solve for the power, x: 108 * 0.8^x = 1; 0.8^x = 1/108; Ln(0.8^x) = ln(1/108); xln(0.8) = ln(1\108); x = ln(1/108) / ln(0.8) = -4.682 / -0.223 = 21 bounces
a) The formula for the height of the ball on the nth bounce is given by h(n) = 108 × (0.8)ⁿ.
b) The height on the 10th bounce is approximately 11.6 inches.
c) The ball will bounce no higher than an inch after around 21 bounces.
Part (a): Formula for h(n)
The initial height from which the ball is dropped is 9 feet. Each bounce reaches 80% of the previous height. We need to convert the height to inches:
9 feet = 9 × 12 = 108 inches
The height h(n) after n bounces can be represented by the formula:
h(n) = 108 × (0.8)ⁿ
Part (b): Height on the 10th bounce
To find the height on the 10th bounce, substitute n = 10 into the formula:
h(10) = 108 × (0.8)¹⁰
Using a calculator:
h(10) ≈ 108 × 0.1073741824 ≈ 11.6 inches
Part (c): Number of bounces before height is no higher than an inch
We need to find n such that:
108 × (0.8)ⁿ ≤ 1
Divide both sides by 108:
(0.8)ⁿ ≤ 1/108
Taking the natural logarithm on both sides:
n ln(0.8) ≤ ln(1/108)
n ≤ ln(1/108) / ln(0.8)
Using a calculator:
n ≤ ln(0.009259259) / ln(0.8) ≈ -4.682131227 / -0.223143551 = 21
Therefore, the ball will rise no higher than an inch after approximately 21 bounces.
The formula for the height of the rubber ball after bounce n is h ( n ) = 108 × ( 0.8 ) n . The height on the 10th bounce is approximately 11.6 inches, and it takes 21 bounces before the ball rises no higher than an inch.
;
