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.
V = pi r^2 h
dV/dr = pi h (2 r)
dV/dr = 12 pi r
To find the ratio of change of the volume(v) with respect to the radius(r) of the base, we need to differentiate the volume formula of a right circular cylinder with respect to the radius.
The volume(V) of a right circular cylinder is given by the formula:
V = π * r^2 * h
Where:
V = Volume of the cylinder
π = Pi (approximately 3.14)
r = Radius of the base
h = Height of the cylinder
Now, differentiate the volume formula with respect to the radius:
dV/dr = d(π * r^2 * h) / dr
Using the product rule of differentiation, we can differentiate the formula:
dV/dr = π * d(r^2 * h) / dr
Now, expand the differentiation using the chain rule:
dV/dr = π * (2r * h) / dr
Simplifying the equation, we get:
dV/dr = 2πrh
Since the height (h) is a fixed value of 6 units, we can substitute its value:
dV/dr = 2πr(6)
Therefore, the ratio of change of the volume (V) with respect to the radius (r) is:
dV/dr = 12πr
So, the ratio of change of the volume with respect to the radius is 12πr.
To find the ratio of change of the volume of a right circular cylinder with respect to its radius, we need to differentiate the formula for the volume of a cylinder with respect to the radius.
The formula for the volume of a cylinder is given by:
V = πr²h
Where V is the volume, r is the radius of the base, and h is the height of the cylinder (which is fixed at 6 units in this case).
To differentiate this equation with respect to the radius (r), we can use partial differentiation as the height (h) is constant:
dV/dr = 2πrh
Since we have a fixed height of 6 units, we can substitute this value into the equation:
dV/dr = 2πr(6)
dV/dr = 12πr
So, the ratio of the change in volume (dV) with respect to the change in radius (dr) is:
dV/dr = 12πr
Therefore, the ratio of change of volume (v) with respect to the change in radius (r) of the cylinder is given by:
v = dV/dr = 12πr