Problem 26 - Calculus: Graphical, Numerical, Algebraic

Question

Volume of Sphere What is the rate of change of the volume of a sphere with respect to the radius when the radius is \(r=2\) in.?

Step-by-Step Solution

Answer

The rate of change of the volume of the sphere with respect to its radius, when the radius is 2 inches, is \(16\pi\) cubic inches per inch.

Step 1: Remember the Formula for Volume of a Sphere

The formula for the volume \(V\) of a sphere is \[V =\frac{4}{3}\pi r^3\] where \(r\) is the radius of the sphere.

Step 2: Apply the Derivative

The rate of change of volume with respect to the radius is given by the derivative of the volume formula with respect to \(r\). So, we need to find \(\frac{dV}{dr} = \frac{d}{dr} (\frac{4}{3} \pi r^3) = 4 \pi r^2\).

Step 3: Evaluate the Derivative at Given Radius

Substitute \(r = 2\) into the derivative result \(\frac{dV}{dr} = 4 \pi r^2\) to find the rate of change of the volume at that radius: \(\frac{dV}{dr} = 4 \pi (2)^2 = 16 \pi\) cubic inches per inch.

Key Concepts

Understanding Derivatives

A derivative represents the rate at which a quantity changes relative to another. In calculus, it's a fundamental concept that helps us understand how things change in a very tiny space. For example, if you imagine a car speeding up, the derivative can tell you how fast its speed is increasing at any given moment.

For the volume of a sphere, we calculate the rate of change of its volume with respect to its radius using the derivative. Essentially, we are looking at how much the volume changes when the radius changes slightly.

• The derivative function provides a "slope" or rate of change for a curve at any specific point.
• It's like zooming into the curve to see how steep it is at a tiny section.
• In our case, we explore this for a sphere's volume formula.

Volume of a Sphere

The volume of a sphere is determined by a well-known formula involving its radius, which is half the diameter of the sphere. This geometric shape's volume is calculated with the formula: \[V = \frac{4}{3} \pi r^3\]where \(V\) is the volume and \(r\) is the radius.

This formula is derived by mathematical principles related to the geometry of spheres. Understanding this formula is crucial when exploring how the sphere's volume changes.

• The \(r^3\) in the formula tells us that the volume increases very quickly as the radius increases.
• This cubic relationship is pivotal in determining the rate at which the volume changes.
• The formula ties directly into our exploration of derivatives to find rates of change.

Rate of Change in Volume

The rate of change is a way of quantifying how one quantity, like volume, changes in response to another, such as the radius of a sphere. By using calculus, specifically derivatives, we can find how quickly the volume changes as the radius grows.

In the context of our sphere, we use the derivative found from the volume formula to find this rate:\[\frac{dV}{dr} = 4 \pi r^2\]This tells us that the change in volume is linked to the square of the radius. So, if the radius is 2 inches, the rate at which the volume changes is \(16 \pi\) cubic inches per inch.

• The outcome highlights how changes in the radius greatly impact volume, due to the squared term \(r^2\).
• It allows for precise calculations in applications like physics and engineering.
• This rate of change is pivotal in real-world scenarios, such as filling a spherical tank with fluid.