Exercise 180 Page 558

Let's illustrate a ball bouncing by drawing a diagram. To get the height of the second bounce, we must multiply the height of the first bounce by the ratio 710. Similarly, the height of the third bounce is the product of the second bounce's height and 710.

Examining the diagram, we see that the heights follow a geometric series. 20, 20( 710), 20( 710)^2, 20( 710)^3, 20( 710)^4 As we can see, the initial value is 20, the multiplier is 710, and we have 5 numbers. This information is all we need to calculate the sum of this series with the following formula. S_n=a_1(1-r^n)/1-r By substituting the initial height a_1= 20, the multiplier r= 710, and the number of bounces n= 5 into the formula, we can determine S_n.

S_n=a_1(1-r^n)/1-r

SubstituteValues

Substitute values

S_5=20(1-( 710)^5)/1- 710

▼

Evaluate right-hand side

CalcQuot

Calculate quotient

S_5=20(1-(0.7)^5)/1-0.7

SubTerm

Subtract term

S_5=20(1-(0.7)^5)/0.3

UseCalc

Use a calculator

S_5=55.462

Notice that this sum only describe the distance the ball travels when going up — but what goes up must also come down.

Therefore, we must multiply the distance by 2 to get the full distance it has traveled. 55.462(2)=110.924 feet