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Chapter 6: Problem 28

A conditioning drill consists of repeatedly running from one end of a basketball court to the other, turning around and running back. Sometimes the drill is changed and the runner turns around at half court, or perhaps at three-fourths of the length of the court. Describe how the number of round trips a runner can do each minute changes when the distance is changed and how this is related to a guitarist changing the note generated by a string by pressing a finger on it at some point.

Short Answer

Expert verified
Answer: Both scenarios can be connected in terms of proportionality as changing the length (distance for the runner and string length for the guitarist) leads to the variation in the outcome (number of round trips for the runner and frequency of the note for the guitarist). In each case, a shorter length results in a higher 'output' (more round trips for the runner and higher frequency for the guitarist).

Step by step solution

01

Understand the basketball court situation

Let's consider a standard basketball court with a length of 94 feet. When the runner goes from one end to the other and returns back, it is called a "round trip," and the total distance covered will be 94 feet * 2 = 188 feet. Let's assume the runner can do 'x' round trips per minute on the full court.
02

Calculate round trips for half and three-fourths courts

Now, let's consider the situations when the runner turns around at half court or three-fourths of the court. The distances for half-court and three-fourths court round trips will be as follows: Half-court: 94 feet * 1/2 = 47 feet (94 feet for a round trip) Three-fourths court: 94 feet * 3/4 = 70.5 feet (141 feet for a round trip) Now we need to find the number of round trips per minute for each case.
03

Use proportionality to find the number of round trips

As the runner's speed is constant, we can use proportionality to calculate the number of round trips for half-court and three-fourths court situations. For half-court scenario: If the runner can do 'x' round trips on the full court (188 feet), then in the same time, the runner can do 'y' round trips on half-court (94 feet). Using proportionality, we can write the equation: x/188 = y/94 -> y = (94 * x) / 188 For the three-fourths court scenario: If the runner can do 'x' round trips on the full court (188 feet), then in the same time, the runner can do 'z' round trips on the three-fourths-court (141 feet). Using proportionality, we can write the equation: x/188 = z/141 -> z = (141 * x) / 188
04

Relate the basketball court situation to the guitarist's string

When a guitarist presses a finger on a string at a certain point, it changes the length of the string that can vibrate freely. This is similar to the runner changing the distance between the points where they turn around. The shorter the length of the vibrating string, the higher the frequency of the note. It is analogous to more round trips for shorter distances. In both situations, changing the length (distance for the runner and string length for the guitarist) leads to the variation in the outcome (number of round trips for the runner and frequency of the note for the guitarist). This relationship demonstrates how two seemingly different scenarios can be related in terms of proportionality.

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