Answers:
The new volume is 1/8 of the old volume
The new surface area is 1/4 of the old surface area
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Step-by-step explanation:
Volume:
Let's say the diameter is d = 60
That would make the radius r = d/2 = 60/2 = 30
The volume of the sphere is
V = (4/3)pi*r^3
V = (4/3)*pi*(30)^3
V = (4/3)*(30)^3*pi
V = 36000*pi
Now consider that d = 30 which is half of the original diameter.
So r = d/2 = 30/2 = 15
And also,
V = (4/3)pi*r^3
V = (4/3)*pi*(15)^3
V = (4/3)*(15)^3*pi
V = 4500*pi
The old volume was 36000pi and the new volume is 4500pi
Comparing the two volumes, we see that the reduction factor is
(new volume)/(old volume) = (4500pi)/(36000pi) = 1/8
The new volume is 1/8 of the old volume
smaller volume = (1/8)*(larger volume)
This is equivalent to saying the larger volume is 8 times that of the smaller volume.
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Surface Area:
We'll use the same d and r values from earlier.
The diameter d = 60 leads to r = 30. We'll use this to find the surface area
SA = 4pi*r^2
SA = 4*pi*30^2
SA = 3600pi
For the smaller sphere (d = 30 and r = 15), we have
SA = 4pi*r^2
SA = 4*pi*15^2
SA = 900pi
The reduction factor is (900pi)/(3600pi) = 1/4
The smaller sphere has its surface area 1/4 that of the larger sphere.
smaller surface area= (1/4)*(larger surface area)
This is equivalent to saying the larger surface area is 4 times that of the smaller surface area.