A cylinder’s radius is reduced to 2/5 its original size and the height is quadrupled. How has the volume of the cylinder changed?
oldv = pi r^2 h
newv = pi (2/5 r)^2 (4h) = pi * 4/25 r^2 * 4h = 16/25 pi r^2 h = (16/25) * oldv
To find how the volume of the cylinder has changed, we need to compare the original volume with the new volume.
The formula for the volume of a cylinder is V = πr^2h, where V represents volume, r represents the radius, and h represents the height.
Let's assume the original radius of the cylinder is represented by the variable r1 and the original height by h1.
According to the problem, the radius is reduced to 2/5 its original size. This means the new radius, r2, is calculated by multiplying the original radius by 2/5, which gives us r2 = (2/5)r1.
The height, on the other hand, is quadrupled. This means the new height, h2, is calculated by multiplying the original height by 4, which gives us h2 = 4h1.
Now, let's find the new volume, V2. Substituting r2 and h2 into the volume formula, we get:
V2 = π(r2^2)(h2)
V2 = π((2/5)r1)^2(4h1)
V2 = π(4/25)(r1^2)(4h1)
V2 = (16/25)π(r1^2)(h1)
So, the new volume, V2, is equal to (16/25) of the original volume, V1.
To summarize, the volume of the cylinder is reduced to 16/25 (or 64%) of its original size.