Problem 30 - Algebra 2

Question

Geometry The formula for the volume of a sphere is \(V=\frac{4}{3} \pi r^{3} .\) a. Find the inverse of the formula. Is the inverse a function? b. Use the inverse to find the radius of a sphere that has a volume of \(35,000 \mathrm{ft}^{3}\) .

Step-by-Step Solution

Answer

The inverse function is \(V= \sqrt[3]{\frac{3V}{4\pi}}\) and yes, it's a function. The radius of the sphere with volume 35000 cubic feet, calculated using the inverse function, gives approximately 22.5 feet.

Step 1: Find the Inverse of the Function

To find the inverse of a function, we need to replace the dependent variable with the independent one and solve for it. This means we replace \(V\) with \(r\) and solve for \(V\). So if we have \(V=\frac{4}{3} \pi r^{3}\), swap \(V\) and \(r\) to obtain \(r=\frac{4}{3} \pi V^{3}\). Now, solve for \(V\), which involves getting rid of all the coefficients on the right side through division. After some computation, this results to \(V= \sqrt[3]{\frac{3V}{4\pi}}\).

Step 2: Check if the Inverse is a Function

For the inverse to be a function, each \(x\)-value should have exactly one \(y\)-value. In this case, sphere's volume \(V\) depends solely on its radius \(r\), and each value of \(r\) leads to one and only one value of \(V\), so the inverse is indeed a function.

Step 3: Use the Inverse to Find the Radius

According to the problem, a sphere has a volume of 35000 cubic feet. So, we put this value to the inverse function we found in step 1. This is done by substituting \(V\) with 35000 in \(V= \sqrt[3]{\frac{3V}{4\pi}}\) and computing the result. Thus, \(r= \sqrt[3]{\frac{3 * 35000}{4\pi}}\).

Key Concepts

Volume of a Sphere

Understanding the volume of a sphere is a common need when dealing with three-dimensional shapes. A sphere is a perfectly round object like a ball, and its volume indicates the amount of space it occupies.

The formula for calculating the volume of a sphere is given by: \[ V = \frac{4}{3} \pi r^{3} \] where:

• \( V \) is the volume
• \( \pi \) is a mathematical constant approximately equal to 3.14159
• \( r \) is the radius of the sphere

This equation shows us that the volume relies heavily on the cube of the radius. Even a small change in the radius has a significant impact on the volume. That's because the radius is raised to the power of three.

Function Inverses

The concept of inverse functions is crucial in mathematics. An inverse essentially "undoes" what the original function does. For a given function, the inverse function swaps the dependent and independent variables.

For example, if we start with the formula for the volume of a sphere, \( V = \frac{4}{3} \pi r^{3} \), finding its inverse involves solving for \( r \) in terms of \( V \). To do this, we replace \( V \) with \( r \) and rearrange the equation, eventually solving for \( V \). This new equation allows us to determine the radius given the volume, and is the essence of finding an inverse.

For the given formula of a sphere's volume, the inverse arises from isolating \( r \) to yield \( r = \sqrt[3]{\frac{3V}{4\pi}} \). This process can show direct relationships between two inherently interrelated variables, confirming that the inverse is indeed a function.

Mathematical Solving Steps

Solving mathematical problems step-by-step is like following a recipe. Each step must be understood and executed correctly. This ensures accuracy and comprehension. Finding the inverse of a function is such a procedure.

The solution involves:

• Swapping variables: Replace \( V \) with \( r \) in the formula \( V = \frac{4}{3} \pi r^{3} \).
• Isolating the desired variable: Rearrange to solve for \( V \).
• Confirming the result: Ensure the mathematical operations result in a correct transformation of the original equation.

The calculated inverse of the sphere's volume formula, \( r = \sqrt[3]{\frac{3V}{4\pi}} \), gives us the radius, informing further calculations such as those related to geometry or physics.

Radius Calculation

Once we've found the inverse function, we can use it to calculate specific values. In this context, if we want to find the radius of a sphere given its volume, substituting the volume value into the inverse function gives us the radius.

Suppose we have a sphere with a volume of 35,000 cubic feet. To find its radius, plug the volume into the inverse formula: \[ r = \sqrt[3]{\frac{3 \times 35000}{4\pi}} \] This calculation yields an approximate value for \( r \), providing a straightforward method of determining the radius from a known volume. This example illustrates not only the application of the inverse function but also reinforces the physical relationship between the volume and radius of a sphere.