a right circular cylinder has a fixed height of 6 units. Find the ratio of change of its volume(v) with respect to the radius(r) of its base.
answers key
To find the ratio of change of the volume of a right circular cylinder with respect to the radius of its base, we can use the formula for the volume of a cylinder.
The volume (v) of a right circular cylinder is given by:
v = π * r^2 * h
where:
v = volume of the cylinder
r = radius of the base
h = height of the cylinder
In this case, we are given that the height of the cylinder is fixed at 6 units.
Now, let's take the derivative of the volume with respect to the radius:
dv/dr = d/dx (π * r^2 * 6)
= 6π * 2r
= 12πr
Therefore, the ratio of change of the volume (v) with respect to the radius (r) is:
dv/dr = 12πr
So, the ratio of change is 12πr.
To find the ratio of change of the volume (V) of a right circular cylinder with respect to the radius (r) of its base, we need to differentiate the volume formula with respect to the radius.
The formula for the volume of a right circular cylinder is V = πr²h, where r is the radius of the base and h is the height.
Differentiating the volume formula with respect to r will give us the ratio of change of volume with respect to the radius:
dV/dr = d(πr²h)/dr
Using the power rule of differentiation, where d(xⁿ)/dx = nxⁿ⁻¹, we can differentiate the volume formula:
dV/dr = π * d(r²h)/dr
Now, we can differentiate r²h with respect to r. Since the height (h) is a fixed value of 6 units, which does not change with the radius, the derivative of r²h with respect to r is simply 2rh:
dV/dr = π * 2rh
Since the height (h) is fixed at 6 units, we can substitute this value into the equation:
dV/dr = π * 2r * 6
Simplifying, we get:
dV/dr = 12πr
Therefore, the ratio of change of the volume (V) with respect to the radius (r) of the cylinder's base is 12πr.