If the radius of a cone is cut in half, the volume of the cone is multiplied by what number?
recall that volume of cone is given by
V = pi*(r^2)*h/3
where
pi = 3.14 (constant)
r = radius
h = height
if r becomes (1/2)r,
V,new = pi*[(1/2)r]^2*h/3
V,new = 1/4[pi*(r^2)*h/3]
V,new = (1/4)V
thus multiplied by 1/4.
hope this helps~ :)
original volume = (1/3)π(r^2)h
new volume = (1/3)π(r/2)^2 h = (1/3)π((r^2)/4)h
so multiplied by 1/4
To solve this problem, we need to understand the relationship between the radius and volume of a cone. The formula for the volume of a cone is V = 1/3πr^2h, where V is the volume, r is the radius, and h is the height of the cone.
Now let's consider what happens when we cut the radius in half. If we divide the radius by 2, we get a new radius, let's call it r'. The volume of the new cone with radius r' and height h is given by V' = 1/3π(r'/2)^2h.
To find the ratio between the new volume (V') and the original volume (V), we can calculate V'/V.
V' = 1/3π(r'/2)^2h
= 1/3π(r^2/4)h (since r' = r/2)
Now let's compare this to the original volume V:
V = 1/3πr^2h
To find the ratio V'/V, we can divide V' by V:
V'/V = (1/3π(r^2/4)h) / (1/3πr^2h)
= (1/3π(r^2/4)h) * (3πr^2h/3πr^2h)
= (r^2/4)h / r^2h
= (r^2/4r^2)(h/h)
= 1/4
Therefore, if the radius of a cone is cut in half, the volume of the cone is multiplied by 1/4.