Chapter 11.4, Problem 56PPS

To determine

Find the total distance traveled by the object.

Answer to Problem 56PPS

The total distance traveled by the object is 6 ft.

Explanation of Solution

Given:

An exhibit at a science museum offers visitors the opportunity to experiment with the motion of an object on a spring. One visitor pulls the object down and lets it go. The object travels 1.2 feet upward before heading back the other way. Each time the object changes direction, it decreases its distance by 20% when compared to the previous direction. Find the total distance traveled by the object.

Concept Used:

The object travels 1.2 feet upward before heading back. So $a_1=1.2$

Each time the object changes direction, it decreases its distance by 20% when compared to the previous direction.

So the common ratio is: $r=80\%=0.80$

This is a geometric series. $a_1=1.2\text{and}r=80\%=0.80$

The total number of revolutions done by the tire before it stops spinning is the sum of the series. Because $\left|r\right|=0.80<1$ , we use the formula: $\text{S}=\frac{a_1}{1-r}$

Calculation:

This is a geometric series. $a_1=1.2\text{and}r=80\%=0.80$

we use the formula: $\text{S}=\frac{a_1}{1-r}$

  $\begin{array}{l}{a}_1=1.2;r=0.80;\text{find}\text{S}\\ \text{S}=\frac{a_1}{1-r}=\frac{1.2}{1-0.80}=\frac{1.2}{0.20}=6\end{array}$

The total distance traveled by the object is 6 ft.

Thus, the total distance traveled by the object is 6 ft.