The volume of a right circular cone is given by

V=(1/3)[pie]r^2h, where r is the raius and h is the height
Find (dV/dt) if r=4,h-5,(dr/dt)=6, and (dh/dt)=-3

what is you thinking on the derivative? Use uv formula.

To find (dV/dt), the derivative of the volume with respect to time, we need to use the chain rule. The chain rule states that for a composite function y = g(f(x)), the derivative of y with respect to x is given by dy/dx = (dy/df) * (df/dx).

Given:
V = (1/3)πr^2h
r = 4
h = 5
dr/dt = 6
dh/dt = -3

Let's differentiate V with respect to t by applying the product rule:

dV/dt = (dV/dr) * (dr/dt) + (dV/dh) * (dh/dt)

To find dV/dr, we treat h as a constant and differentiate with respect to r:

dV/dr = (1/3)π * 2rh

Substituting the given values:
dV/dr = (1/3)π * 2(4)(5) = (40/3)π

To find dV/dh, we treat r as a constant and differentiate with respect to h:

dV/dh = (1/3)πr^2

Substituting the given values:
dV/dh = (1/3)π(4)^2 = (16/3)π

Now, let's substitute the calculated values back into the original equation for dV/dt:

dV/dt = (40/3)π * 6 + (16/3)π * (-3)
= (240/3)π - (48/3)π
= (192/3)π
= 64π

Therefore, (dV/dt) is equal to 64π.

To find dV/dt, which represents the rate of change of the volume with respect to time, we need to use the chain rule of differentiation. The chain rule states that if we have a function of multiple variables, we can differentiate it by differentiating each variable with respect to another variable and then multiplying them together.

In this case, we have V, which depends on r and h, both of which can change with time. So, we need to find dV/dt by differentiating V with respect to r (dV/dr) and h (dV/dh), and then multiplying them by dr/dt and dh/dt respectively.

Given that r = 4, h = 5, dr/dt = 6, and dh/dt = -3, let's start by finding dV/dr and dV/dh.

dV/dr: To find dV/dr, we treat h as a constant and differentiate V with respect to r.
dV/dr = (1/3)π * (2r) * h
= (2π/3) * r * h

dV/dh: To find dV/dh, we treat r as a constant and differentiate V with respect to h.
dV/dh = (1/3)π * r^2 * 1
= (π/3) * r^2

Now that we have dV/dr and dV/dh, we can find dV/dt.

dV/dt = (dV/dr) * (dr/dt) + (dV/dh) * (dh/dt)
= [(2π/3) * r * h] * (6) + [(π/3) * r^2] * (-3)
= (4π * r * h) + (-3π * r^2)

Substituting the given values r = 4 and h = 5 into the equation, we can calculate dV/dt.

dV/dt = (4π * 4 * 5) + (-3π * 4^2)
= (80π) + (-48π)
= 32π

Therefore, the rate of change of the volume is 32π.