A balloon is composed of a right circular cone joined at its base to a hemisphere. The diameter of the base is equal to the height of the cone. If the balloon is inflated, at what rate is the volume, V, changing with respect to the height, h.
Volume=1/3 area base * height
= 1/3 PI* (h/2)^2 * h
check that. now do dV/dh
To find the rate at which the volume, V, is changing with respect to the height, h, we need to find the derivative dV/dh.
First, let's simplify the formula for the volume of the balloon:
V = (1/3) * π * (h/2)^2 * h
Now, let's differentiate the volume formula with respect to h using the power rule and the chain rule:
dV/dh = (1/3) * π * [2 * (h/2) * (1/2)] * h + (1/3) * π * (h/2)^2 * 1
Simplifying further:
dV/dh = (1/3) * π * (h/2) * h + (1/3) * π * (h/2)^2
dV/dh = (1/6) * π * h^2 + (1/12) * π * h^2
Combining the terms:
dV/dh = (3/12) * π * h^2 + (2/12) * π * h^2
dV/dh = (5/12) * π * h^2
Therefore, the rate of change of the volume with respect to the height is given by:
dV/dh = (5/12) * π * h^2
So, the volume is changing at a rate of (5/12) * π * h^2 per unit change in height.