The volume V of a sphere is given by the formula:
V = (4/3)πr³
where r is the radius of the sphere, and π (pi) is a constant approximately equal to 3.14159.
To find the rate of change of the volume with respect to the radius, we need to calculate the derivative of the volume formula with respect to r.
Using calculus, the derivative of V with respect to r is:
dV/dr = d/dr [(4/3)πr³]
Applying the power rule of differentiation, we get:
dV/dr = (4/3)π * 3r²
Simplifying this expression yields:
dV/dr = 4πr²
This result means that the rate of change of the volume of the sphere increases with the square of its radius. In other words, as the radius of the sphere grows, the volume increases significantly, and the relationship is quadratic in nature.
For example, if you were to double the radius of the sphere, the volume would grow by a factor of four, demonstrating how sensitive the volume is to changes in radius.
In summary, the rate of change of the volume of a sphere with respect to its radius is:
dV/dr = 4πr²
This shows that for a small increase in radius, the volume increases by this amount, and it highlights the geometric nature of the sphere’s volume in relation to its dimensions.