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) .

Question

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) .

Answer 1

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.

Answer 2

To find the value of a, we need to differentiate the volume formula for a sphere with respect to its radius.

Given:
Volume of a sphere, V = (4/3)πr^3

To find dV/dr, we differentiate both sides of the equation with respect to r:
dV/dr = d/dx (4/3)πr^3

Using the power rule of differentiation, where d/dx (x^n) = nx^(n-1), the derivative of r^3 with respect to r is:
dV/dr = (4/3)π * 3r^2

Simplifying further:
dV/dr = 4πr^2

Comparing this to dV/dr = aVr, we can see that a = 4π.

Hence, the value of a is 4π.

Answer 3

To find the instantaneous rate of change of the volume of a sphere, we can take the derivative of the volume formula with respect to the radius.

The volume formula for a sphere is given by:

V = (4/3)πr^3

To find dV/dr, we can differentiate both sides of the equation with respect to r:

dV/dr = d/dx[(4/3)πr^3]

Now we can apply the power rule of differentiation, which states that the derivative of x^n, where n is a constant, is n*x^(n-1). Using this rule, we can find the derivative of the right side of the equation:

dV/dr = (4/3)π * d/dr [r^3]
= (4/3)π * 3r^2
= 4πr^2

So, the instantaneous rate of change of the volume of a sphere with respect to its radius is given by dV/dr = 4πr^2.

Now let's compare this equation to the given equation dV/dr = aV/r:

We can substitute the volume formula (V = (4/3)πr^3) into the equation:

4πr^2 = a*(4/3)πr^3 / r

Simplifying:

4πr^2 = (4/3)πar^2

Comparing the coefficients, we find:

4 = (4/3)a

To solve for a, we can multiply both sides of the equation by (3/4):

(3/4)*4 = (3/4)*(4/3)a

3 = a

Therefore, the value of a is 3.