Find the instantaneous rate of change of the volume V=(1/3) pi r^2 H (all put together) of a cone with respect to the radius r at r=a if the height H does not change.
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Differentiate with respect to r ONLY
dV/dr = (2/3)pi r H @ r = a
= (2/3) pi a H
Usually when you talk about rates it is with respect to time. If you knew the rate that r was increasing, you could multiply it by dV/dr above to get that rate.
To find the instantaneous rate of change of the volume V of the cone with respect to the radius r at r = a, we can use calculus.
1. Start with the formula for the volume of a cone: V = (1/3)πr²H, where r is the radius and H is the height.
2. Differentiate both sides of the equation with respect to r. This will give us the rate of change of V with respect to r.
dV/dr = (1/3)π(2r)H
3. Simplify the equation:
dV/dr = (2/3)πrH
This equation represents the rate of change of V with respect to r for any value of r.
4. Substitute r = a into the equation to find the instantaneous rate of change at r = a:
Instantaneous rate of change at r = a = (2/3)πaH
So, the instantaneous rate of change of the volume of the cone with respect to the radius at r = a is (2/3)πaH.
To find the instantaneous rate of change of the volume of a cone with respect to the radius when the height is constant, we need to use calculus. We will differentiate the volume function V with respect to the radius r and then evaluate it at r = a.
The volume of a cone can be given by the formula V = (1/3) πr^2H, where r is the radius, H is the height, and π is a constant (approximately equal to 3.14159).
Let's find the derivative of V with respect to r:
dV/dr = d/dt [(1/3) πr^2H] (since H is constant with respect to r)
= (1/3)π * d/dr (r^2H)
= (1/3)π * (2rH) (using the power rule of derivatives, d/dx (x^n) = nx^(n-1))
Simplifying, we get:
dV/dr = (2/3)πrH
Now, to find the instantaneous rate of change of the volume with respect to the radius at r = a, we substitute a for r in the expression we just obtained:
Instantaneous rate of change at r = a = (2/3)πaH
Thus, the instantaneous rate of change of the volume of the cone with respect to the radius at r = a is (2/3)πaH.