The volume of a cone is calculated using the formula: V = (1/3)πr²h, where r stands for the radius and h for the height of the cone.
When both the radius and the height are increasing at a constant rate of 12 centimeters per second, we can differentiate the volume with respect to time to find out how the volume changes.
Using the product rule for differentiation, we get:
dV/dt = (dV/dr)(dr/dt) + (dV/dh)(dh/dt)
First, we need to calculate the partial derivatives:
- dV/dr = (2/3)πr * h
- dV/dh = (1/3)πr²
Substituting these into the equation gives us:
dV/dt = (2/3)πr * h * (dr/dt) + (1/3)πr² * (dh/dt)
Given that both dr/dt and dh/dt are 12 centimeters per second, we substitute these values into the equation:
dV/dt = (2/3)πr * h * 12 + (1/3)πr² * 12
Factoring out common terms, we get:
dV/dt = 4π(2rh + r²) cm³/s
This equation shows how the volume of the cone (V) changes over time as both the radius (r) and height (h) increase. The rate of volume change will depend on the specific values of r and h at any given moment.