A monkey is swinging from a tree. On the first swing, she passes through an arc whose length is 24 m 24 m24, start text, space,
2 answers:
Answer:
Step-by-step explanation:
<u>Complete Question:</u>
A monkey is swinging from a tree. On the first swing, she passes through an arc whose length is 24 m. With each swing, she travels along an arc that is half as long as the arc of the previous swing. Which expression gives the total length the monkey swings in her first n swings?
<u>Solution:</u>
First Swing:
24 m
Second Swing:
Half of first, so
24(1/2)
Third Swing:
Half of second, so:
24(1/2)(1/2)
Fourth Swing:
24(1/2)(1/2)(1/2)
So we see the general formula as:
<u>Note:</u> n -1 because (1/2) plays a role from second swing, which is n - 1
where n is the number of swings
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Parametrization of the first segment:
x=t+1, y=4t, z=1 wherein t is on the segment [0,1].
Second segment:
x=2, y=2t+4, z=2t+1 and again t is in [0,1].
Compute the derivatives like this:
First segment: dx=1dt, dy=4dt, dz=0
Second segment:dx=0, dy=2dt, dz=2dt.
Using the above variables, the given integral becomes like this:
The above integral is classical, and simple computation we obtain:
x=t+1, y=4t, z=1 wherein t is on the segment [0,1].
Second segment:
x=2, y=2t+4, z=2t+1 and again t is in [0,1].
Compute the derivatives like this:
First segment: dx=1dt, dy=4dt, dz=0
Second segment:dx=0, dy=2dt, dz=2dt.
Using the above variables, the given integral becomes like this:
The above integral is classical, and simple computation we obtain: