the volume of a cone is 48 pi cubic units. If its dimensions are reduced to one half their original length, what would its new volume be?
v = 1/3 pi * r^2 * h
if dimensions are multiplied by 1/2, we get
1/3 * pi * (r/2)^2 * (h/2)
= 1/8 * 1/3 * pi * r^2 * h
= 1/8 v
or, 1/8 = (1/2)^3 the previous volume
just FYI, the area would be (1/2)^2 = 1/4 the previous area.
To find the new volume of a cone when its dimensions are reduced by one-half, we need to follow these steps:
Step 1: Find the original volume of the cone.
Given that the volume of the cone is 48π cubic units, we have:
Volume = 48π cubic units
Step 2: Reduce the dimensions to one-half.
When the dimensions are reduced to one-half, both the height and the radius will be halved.
Step 3: Use the formula for the volume of a cone.
The formula for the volume of a cone is:
Volume = (1/3)πr^2h
where r is the radius of the base and h is the height.
Step 4: Calculate the new volume.
Since the dimensions have been reduced by one-half, we can substitute the new values into the volume formula:
New Volume = (1/3)π (r/2)^2 (h/2)
Simplify the equation:
New Volume = (1/3)π (r^2/4) (h/2)
= (1/3)π (r^2*h/8)
= (1/3) * (1/8) * πr^2h
Notice that the new volume is (1/3) * (1/8) of the original volume. Therefore, the new volume is:
New Volume = (1/3) * (1/8) * 48π cubic units
= 4π cubic units
So, the new volume of the cone, when its dimensions are reduced to one-half, is 4π cubic units.