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A ball is dropped from a height of 15 feet and bounces. Suppose that each bounce is 5/8 of the height of the bounce before. Thus, after the ball hits the floor for the first time, it rises to a height of 15(g) = 9.375 feet, etc. (Assume g = 32ft/s² and no air resistance.) A. Find an expression for the height, in feet, to which the ball rises after it hits the floor for the nth time: hn = B. Find an expression for the total vertical distance the ball has traveled, in feet, when it hits the floor for the first, second, third and fourth times: first time: D = second time: D = third time: D= fourth time: D= C. Find an expression, in closed form, for the total vertical distance the ball has traveled when it hits the floor for the nth time. Dn=

A ball is dropped from a height of 15 feet and bounces. Suppose that each bounce is 5/8 of the height of the bounce before. Thus, after the ball hits the floor for the first time, it rises to a height of 15(g) = 9.375 feet, etc. (Assume g = 32ft/s² and no air resistance.) A. Find an expression for the height, in feet, to which the ball rises after it hits the floor for the nth time: hn = B. Find an expression for the total vertical distance the ball has traveled, in feet, when it hits the floor for the first, second, third and fourth times: first time: D = second time: D = third time: D= fourth time: D= C. Find an expression, in closed form, for the total vertical distance the ball has traveled when it hits the floor for the nth time. Dn=

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Problem Statement: A ball is dropped from a height of 15 feet and bounces. Suppose that each bounce is 5/8 of the height of the bounce before. Thus, after the ball hits the floor for the first time, it rises to a height of \(15 \left( \frac{5}{8} \right) = 9.375\) feet, etc. (Assume \( g = 32 \, \text{ft/s}^2 \) and no air resistance.) ### Part A: **Find an expression for the height, in feet, to which the ball rises after it hits the floor for the \( n \)th time:** \[ h_n = \] ### Part B: **Find an expression for the total vertical distance the ball has traveled, in feet, when it hits the floor for the first, second, third, and fourth times:** - First time: \( D = \) \[ \] - Second time: \( D = \) \[ \] - Third time: \( D = \) \[ \] - Fourth time: \( D = \) \[ \] ### Part C: **Find an expression, in closed form, for the total vertical distance the ball has traveled when it hits the floor for the \( n \)th time.** \[ D_n = \] This problem involves understanding and applying the principles of geometric sequences and series. Each bounce height is a fraction of the previous one, and the total distance traveled involves summing a series of these heights along with the distances fallen.
Transcribed Image Text:### Problem Statement: A ball is dropped from a height of 15 feet and bounces. Suppose that each bounce is 5/8 of the height of the bounce before. Thus, after the ball hits the floor for the first time, it rises to a height of \(15 \left( \frac{5}{8} \right) = 9.375\) feet, etc. (Assume \( g = 32 \, \text{ft/s}^2 \) and no air resistance.) ### Part A: **Find an expression for the height, in feet, to which the ball rises after it hits the floor for the \( n \)th time:** \[ h_n = \] ### Part B: **Find an expression for the total vertical distance the ball has traveled, in feet, when it hits the floor for the first, second, third, and fourth times:** - First time: \( D = \) \[ \] - Second time: \( D = \) \[ \] - Third time: \( D = \) \[ \] - Fourth time: \( D = \) \[ \] ### Part C: **Find an expression, in closed form, for the total vertical distance the ball has traveled when it hits the floor for the \( n \)th time.** \[ D_n = \] This problem involves understanding and applying the principles of geometric sequences and series. Each bounce height is a fraction of the previous one, and the total distance traveled involves summing a series of these heights along with the distances fallen.
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