To find a, we need to take the derivative of the volume formula with respect to the radius, dV/dr.
First, let's simplify the volume formula:
V = (4/3)πr^3
Now, let's take the derivative of V with respect to r:
dV/dr = (d/dx)[(4/3)πr^3]
= (4/3)π(3r^2)
= 4πr^2
We now have the derivative of V with respect to r.
This derivative represents the rate of change of the volume of the sphere with respect to its radius.
So, we can express the instantaneous rate of change of V with respect to r as dV/dr = 4πr^2.
Comparing this with dV/dr = aV/r, we can conclude that a = 4πr^2.
The instantaneous rate of change of the volume V of a sphere with respect to its radius r can be expressed as dVdr=aVr . What is a? Recall that the volume formula for a sphere is V=43πr3 . Show all your work. If limit notation is needed in your response, use the following example as a guide: limx→1(x2+2) should be written as lim x→1 (x2+2) .
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