Question

30 pointsss

At the neighborhood playground, your friend sits on a swing and gets pulled back by you to the highest point that you can reach. When you let go, your friend rides the swing without receiving or providing additional push. Your friend travels along an arc of 14 feet on the first swing (back to front). As the swing comes backwards, the length of the arc is .80 of the previous swing. This continues: each successive arc created by swinging is .80 of the previous swing.
a. Find the length of the arc for the 5th swing (the 5th arc). Round to the thousandths place.
b. On which swing is the length of the arc first less than 1 foot?
c. Find the total distance your friend travels in all of the swings up to and including the swing in answer to letter b.

Answer 1

• 5th swing: 5.734 ft
• 13 swing is less than 1 ft
• 66.152 feet traveled in 13 swings

Explanation:

The sequence of arc lengths is a geometric sequence with first term 14 ft and common ratio 0.8. The general (n-th) term of such a sequence is given by ...

an = a1 · r^(n-1) . . . . . . . first term a1, common ratio r

For this scenario, the n-th term is ...

an = 14·0.8^(n-1)

__

a.

The 5th term is ...

a5 = 14·0.8^(5-1) ≈ 5.734 . . . . feet

__

b.

For the arc length to be less than 1 ft, we require ...

14·0.8^(n-1) < 1

0.8^(n-1) < 1/14

(n -1)log(0.8) < log(1/14) . . . . . . note that these log values are negative

n -1 > log(1/14)/log(0.8)

n > 1 +log(1/14)/log(0.8) ≈ 12.8

The 13th swing will have an arc length less than 1 ft.

__

c.

The sum of n terms of a geometric sequence is given by ...

Sn = a1 · (1 -r^n)/(1 -r)

13 terms of our sequence will total ...

S13 = 14 · (1 -0.8^(13))/(1 -0.8) ≈ 66.152 . . . feet

The total distance traveled in 13 swings is about 66.152 feet.