Question

pendulum swings through an arc of 50 cm. On each successive swing, the length of the arc is 85% of the previous length. a. Find the total distance that the pendulum would have swung after 7 swings. b. When it stops, what total distance will the pendulum have swung?

Answer 1

Answer:

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Step-by-step explanation:

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: an=a1*rn-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 an=54*0.927-1=32.743 cm;

b)we can find it by inequality an=a1*rn-1<17,

54*0.92n-1<17,

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

ln0.92n-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;

Sn=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 ;

Sn=a1/(1-r)=54/0.08=675 cm

THIS IS THE ANSWER