Answer:
The answers for your two questions are the following
a)11.606 m
b) As the number of swings approaches infinity
Step-by-step explanation:
We are dealing with an exponential equation series, where the length of each swing can be represented as
l = [ 15 *(0.95)^(n-1) ]
n is the corresponding number of the swing.
So, for the first swing, n = 1
[ 15 *(0.95)^(1-1) ] = 15 m
a) How far with the pendulum travel on its 6th swing?
We just need to evaluate the previous formula for n = 6
[ 15 *(0.95)^(6-1) ] =
[ 15 *(0.95)^(5) ] =
[ 15 *(0.7737) ] =
[ 15 *(0.7737) ] = 11.606 m
b) How far will the pendulum swing before it essentially stops? Hint: This is an infinite geometric series.
We previously stated that the length of the arc of each swing can be represented as
l(n) = [ 15 *(0.95)^(n-1) ] , for n>=1
Since the function approaches zero if and only if n approaches infinity, we can say that the pendulum never stops.
Of course, this only happens mathematically, we can always fin a threshold for which the movement cannot be registered anymore.
Please see attached graph for a representation of the function
Answer:
a) It will travel approx 79.47 meters,
b) It will travel 300 meters.
Step-by-step explanation:
Given,
The initial distance travel on first swing = 15 meters,
Also, Each subsequent swing is 95% of the previous swing.
Thus, there is a G.P. that shows this situation,
Having first term, a = 15,
And, the common difference, r = 95 % = 0.95,
a) Also, for the 6th swing,
Number of terms, n = 6,
Hence, the distance covered by the pendulum on its 6th swing,
[tex]S_{n}=\frac{a(1-r^n)}{1-r}[/tex]
[tex]S_{6}=\frac{15(1-0.95^6)}{1-0.95}[/tex]
[tex]=79.4724328125\approx 79.47\text{ meters}[/tex]
b) When [tex]n=\infty[/tex]
The distance will the pendulum swing before it essentially stops is,
[tex]S_{\infty}=\frac{a}{1-r}[/tex]
[tex]=\frac{15}{1-0.95}[/tex]
[tex]=300\text{ meters}[/tex]
