Answer to Question #147937 in Calculus for Nur atikah

Question

A pendulum swing through an arc of length 54cm. On each successive swing, the length
of the arc is 0.92 of the previous length.
(a) What is the length of the arc after 7 swings.
(b) At which swing is the length of the arc of the pendulum less than 17cm for the
first time.
(c) Find the total distance covered by the pendulum after 22 swings.
(d) Find the total distance covered by the pendulum before it comes to a stop.

Answer 1

we have initial length of the arc, S=54cm and each next length is 0.92 times shorter than the previous.

so we can take it as geometric series;

a) the length of the arc after 7 swings can be calculated by the formula: a_n=a₁*r^n-1

n is the order of the term, a₁ is the first term, r is in our case equal to 0.92;

so we have a_n=54*0.92^7-1=32.743 cm;

b)we can find it by inequality a_n=a₁*r^n-1<17,

54*0.92^n-1<17,

0.92^n-1<17/54; we put ln on both sides

ln0.92^n-1<ln(0.315);

(n-1)ln0.92<ln(0.315);

we calculate logarithmic values by calculator;

(n-1)-0.0834<-1.155; we multiply both side by -1 and open the brackets;

0.0834n-0.0834>1.155;

from here we can find n>14.85, n=15, after 15 swings the length of the arc will be less than 17 cm;

c) the distance after 22 swings can be calculated by the formula of the sum of geometric series;

S_n=a₁*(rⁿ-1)/(r-1) the sum equals to S₂₂=54*(0.92²²-1)/(0.92-1)=567.19 cm;

d) total distance can be calculated by the formula of the sum absolute decreasing geometric series ;

S_n=a₁/(1-r)=54/0.08=675 cm