Title of project: pendulum waves. Pendulums Waves Pendulums Waves.
EXAMPLE 4 Use an infinite series as a model Pendulums A pendulum that is released to swing freely...
7
EXAMPLE 4 Use an infinite series as a model Pendulums A pendulum that is released to swing freely travels 18 inches on the first swing. On each successive swing, the pendulum travels 80% of the distance of the previous swing. What is the total distance the pendulum swings?
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Transcript of EXAMPLE 4 Use an infinite series as a model Pendulums A pendulum that is released to swing freely...
EXAMPLE 4 Use an infinite series as a model
Pendulums
A pendulum that is released to swing freely travels 18 inches on the first swing. On each successive swing, the pendulum travels 80% of the distance of the previous swing. What is the total distance the pendulum swings?
EXAMPLE 4 Use an infinite series as a model
The total distance traveled by the pendulum is:
d = 18 + 18(0.8) + 18(0.8)2 + 18(0.8)3 + · · ·
a1
1 – r=
= 90
181 – 0.8=
Write formula for sum.
Substitute 18 for a1 and 0.8 for r.
Simplify.
The pendulum travels a total distance of 90 inches, or 7.5 feet.
ANSWER
SOLUTION
EXAMPLE 5 Write a repeating decimal as a fraction
Write 0.242424. . . as a fraction in lowest terms.
0.242424. . . = 24(0.01) + 24(0.01)2 + 24(0.01)3 + · · ·a1
1 – r=
24(0.01)1 – 0.01=
0.240.99
=
2499=
833=
Write formula for sum.
Substitute 24(0.01) for a1 and 0.01 for r.
Simplify.
Write as a quotient of integers.
Reduce fraction to lowest terms.
The repeating decimal 0.242424. . . is 833 as a fraction.
ANSWER
GUIDED PRACTICE for Examples 4 and 5
5. WHAT IF? In Example 4, suppose the pendulum travels 10 inches on its first swing. What is the total distance the pendulum swings?
SOLUTION
The total distance traveled by the pendulum is:
d = 10 + 10(0.8) + 10(0.8)2 + 10(0.8)3 + · · ·
a1
1 – r=
101 – 0.8= = 10
0.2 = 50 inches
GUIDED PRACTICE for Examples 4 and 5
Write the repeating decimal as a fraction in lowest terms.
6. 0.555. . .
SOLUTION
0.555. . . = 55(0.01) + 55(0.01)2 + 55(0.01)3 + · · ·
a1
1 – r=
55(0.01)1 – 0.01
= 0.550.99
= 59=
GUIDED PRACTICE for Examples 4 and 5
Write the repeating decimal as a fraction in lowest terms.
7. 0.727272. . .
SOLUTION
0.727272. . . = 72(0.01) + 72(0.01)2 + 72(0.01)3 + · · ·
a1
1 – r=
0.720.99=
72(0.01)1 – 0.01=
= 7299
811=
GUIDED PRACTICE for Examples 4 and 5
Write the repeating decimal as a fraction in lowest terms.
8. 0.131313. . .
SOLUTION
0.131313. . . = 13(0.01) + 13(0.01)2 + 13(0.01)3 + · · ·
a1
1 – r=
0.130.99=
13(0.01)1 – 0.01=
= 1399
