/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 81 In this set of exercises, you wi... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 81

In this set of exercises, you will use sequences and their sums to study real- world problems. A ball dropped to the floor from a height of 10 feet bounces back up to a point that is three-fourths as high. If the ball continues to bounce up and down, and if after each bounce it reaches a point that is three-fourths as high as the point reached on the previous bounce, calculate the total distance the ball travels from the time it is dropped to the time it hits the floor for the third time.

Short Answer

Expert verified
The total distance the ball travels from the time it is dropped to the time it hits the floor for the third time is approximately 49.43 feet.

Step by step solution

01

Understand the Sequence

The ball initially falls from a height of 10ft then bounces three times. Each bounce reaches a height that is three-fourths (or 0.75) the previous height. Formulate this sequence as: 10, 10*0.75, 10*0.75^2, 10*0.75^3, 10*0.75^4.
02

Consider the Total travel distance

The total distance, D the ball traveled should consider each bounce which consists of a fall and rise (except for the first fall and final drop). Therefore, the distance can be computed from our sequence as: D = 10 + 2*(10*0.75) + 2*(10*0.75^2) + 2*(10*0.75^3) + 10*0.75^4.
03

Calculate the Distance

Calculate the total travel distance D using above expression: D = 10 + 2*(7.5) + 2*(5.625) + 2*(4.21875) + 3.1640625. Summing these up, we find that D will approximately be 49.43ft.
04

Validate

Ensure that the number of terms and values used align with the description in the question. There should be 5 terms in total (representing the initial fall and 3 bounces), and the reduction factor 0.75 is applied at each bounce.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Geometric Sequences
A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous term by a constant, called the common ratio. In this exercise, the sequence describes the heights of each bounce of the ball.
Starting at a height of 10 feet, each subsequent bounce reaches 75% of the previous height. We express this as a geometric sequence: 10, 10 \( \times 0.75 \), 10 \( \times 0.75^2 \), and so on. Each term gets progressively smaller as the ball's bounces diminish.
This sequence is important as it helps calculate the distance the ball travels each time it falls and rises. Understanding how the sequence shrinks with each term is crucial to solve this problem.
Series
A series is the sum of the terms of a sequence. In this problem, we are interested in the series because it represents the total distance the ball travels.
Our objective is to find the sum of all distances traveled by the ball, from the initial drop to reaching the floor after each bounce. We know:
  • The initial fall: 10 feet
  • Each subsequent bounce up and down after the first is doubled: 2*(10 \( \times 0.75 \)), 2*(10 \( \times 0.75^2 \)), etc.
We sum these to get the total distance. This series essentially captures the repetitive nature of the bounces: a downward and an upward motion.
Understanding how to form this series appropriately ensures the correct computation of the ball's total travel distance.
Problem Solving
Problem solving involves applying various mathematical concepts to reach a solution. In this scenario, we started by identifying the nature of the sequence describing the ball's bounces and translating it into a series to compute total distance.
To solve the problem:
  • Identify what you know: the initial height, the ratio, and number of bounces.
  • Set up the mathematical model: a geometric sequence and its resulting series.
  • Compute the solution: using the series to calculate total distance.
This structured approach helps break down complex problems into manageable steps. By systematically following these steps, it becomes easier to solve similar sequence and series-related problems in various contexts.

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Most popular questions from this chapter

In this set of exercises, you will use sequences to study real-world problems. Knitting New trends in knitting involve creating vibrant patterns with geometric shapes. Suppose you want to knit a large right triangle. You start with 85 stitches and decrease each row thereafter by 2 stitches. (a) What type of sequence does the number of stitches in each row produce: arithmetic, geometric, or neither? (b) Find a rule that gives the number of stitches for the nth row. (c) How many rows must be knitted to end with a row of just one stitch?

A diagonal of a polygon is defined as a line segment with endpoints at a pair of nonadjacent vertices of the polygon. How many diagonals does a pentagon have? an octagon? an \(n\) -gon (that is, a polygon with \(n\) sides)?

In a telephone survey, people are asked whether they have seen each of four different films. Their answers for each film (yes or no) are recorded. (a) What is the sample space? (b) What is the probability that a respondent has seen exactly two of the four films? (c) Assuming that all outcomes are equally likely, what is the probability that a respondent has seen all four films?

Concepts This set of exercises will draw on the ideas presented in this section and your general math background. \- A sequence \(b_{0}, b_{1}, b_{2}, \ldots\) has the property that \(b_{n}=\) \(\left(\frac{n+3}{n+2}\right) b_{n-1}\) for \(n=1,2,3, \ldots,\) where \(c\) is a positive constant to be determined. Find \(c\) if \(b_{2}=25\) and \(b_{4}=315\)

Involve dialing the last four digits of a phone number that has an area code of 907 and an exchange of \(316 .\) The exchange consists of the first three digits of the seven-digit phone number. How many outcomes are there for dialing the last four digits of a phone number?
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