M09-25

Question

If the Earth's orbit around the Sun is a circle, by how much will the length of the Earth's orbit increase if the radius of this orbit grows by  \frac{\pi}{2}  meters?

A. 1 meter
B. 2 meters
C.  \pi  meters
D.  2\pi  meters
E.  \pi^2  meters

Bunuel · Answer

Official Solution:

If the Earth's orbit around the Sun is a circle, by how much will the length of the Earth's orbit increase if the radius of this orbit grows by  \frac{\pi}{2}  meters?

A. 1 meter
B. 2 meters
C.  \pi  meters
D.  2\pi  meters
E.  \pi^2  meters

Let  R  be the radius of the orbit. The difference in length between the current orbit and the hypothetical one will be  2\pi(R + \frac{\pi}{2}) - 2 \pi R = 2 \pi \frac{\pi}{2} = \pi^2

Answer: E

SQUINGEL · Answer

i am lost..............please help

msarji · Answer

I don't quite follow the explanation given either. However, I think a simpler way to solve may be just to realize that orbit=circumference. So if the radius increases by r/2 how much does the circumference increase. And since circumference= pi*di if the radius increases by pi/2 multiply by 2 to get diameter=pi, and then plug into circumference equation you get pi*pi or pi^2.

dorey777 · Answer

Details of Bunuel's method:

The length of the orbit is asking for the circumference.

If the new orbit (circle) is increased by π/2 then the circumference of this would be 2π(R + π/2). The question asks for the increase (or the difference) of the new circumference to the original circumference. Thus 2π(R + π/2) - 2πR. When using distributives it would = 2πR + 2π^2 / 2 - 2πR. The 2π^2 /2 cancels out to be π^2 and 2πR - 2πR cancels out. Thus π^2 becomes the answer.

Hope this helps.

ChrisL1988 · Answer

You can also pick smart numbers:

r1=π 
 c1=2rπ=2π^2

r2=\frac{3π}{2} 
 c2=2rπ=2\frac{3π}{2}π=3π^2

diff=3π^2-2π^2=π^2

badly777 · Answer

Easy breezy word problem with some geometry

Lenght of the orbit is calculated by = 2pi*r (circumference)
Then you just have to add the formula above pi/2 which will give

2*pi*(r+pi/2) = 2pi*r+pi^2

So, pi^2 is the answer

aarushi1710 · Answer

Initial length of the earth's orbit (Let this be P1) = Perimeter of a circle

P1 = 2πr

We want to know the new length of the earths orbit (new perimeter, say P2) if the radius increases BY π/2

P2 = 2π(r+π/2)

P2 = 2πr + 2π^2/2

P2 = 2πr + π^2

P2 = P1 + π^2

Hence, earths orbit increases by π^2 meters

Answer choice (E)

M09-25

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