Chapter 11.3, Problem 46PPS

To determine

Find the percent of the original amount of dirt is picked up after the seventh time the vacuum is run over the carpet.

Answer to Problem 46PPS

99.99%

Explanation of Solution

Given:

A vacuum claims to pick up 80% of the dirt every time it is run over the carpet. Assuming this is true, what percent of the original amount of dirt is picked up after the seventh time the vacuum is run over the carpet?

Concept Used:

After the first use of the vacuum, pick up 80% or 0.80 from the carpet is cleaned and it will remain 1 − 0.80 = 0.20 still dirty.

After the 2^nd use of the vacuum, the still dirty part will be:

  $0.20-0.80·0.20=0.20·(1-0.80)={0.20}^2$

Calculation:

Thus a geometric sequence models the problem. We consider $a_n$ the fraction of the carpet which remained still; dirty after the nth use of the vacuum: $a_1=0.20;r=0.20$

The term $a_7$ represents the fraction of the carpet remained still dirty after the 7^th use of the vacuum.

  $a_7=a_1·r^6=0.20·{(0.20)}^6={0.20}^7=0.0000128$

The percent of dirt picked up after the 7^th use is: $1-a_7=1-0.0000128=0.9999872\approx 99.99\%$

Thus, 99.99% of the original amount of dirt is picked up after the seventh time the vacuum is run over the carpet.