Question

All six sides of a cubical metal box are \(0.25\) inch thick, and the volume of the interior of the box is 40 cubic inches. Use differentials to find the approximate volume of metal used to make the box.

Short answer

The approximate volume of metal used is found using differentials, considering the small volume change due to metal thickness.

Step-by-step solution

Understand the Problem

We have a cube with a given interior volume of 40 cubic inches. Each side of the cube is made of metal and has a thickness of 0.25 inches. Our task is to find the volume of the metal used to construct the cube.

Calculate the Interior Side Length

Since the volume of the interior of the cube is 40 cubic inches, we need to find the side length of the cube excluding the metal. The formula for the volume of a cube is given by the side length cubed: \[ s_i^3 = 40 \] where \( s_i \) is the interior side length of the cube. Taking the cube root gives us \( s_i = \sqrt[3]{40} \).

Calculate the Exterior Side Length

The exterior side length of the cube takes into account the metal thickness. Each side is thickened by 0.25 inches, so the total addition due to thickness is 0.5 inches (0.25 inches from each side). Hence, the exterior side length is calculated as follows:\[ s_e = s_i + 0.5 \]

Apply Differentials to Find Volume of Metal

The volume of the entire cube (including metal) is given by \( s_e^3 \). The metal volume can be approximated as the difference in volumes, which for small changes can be represented using differentials: \[ dV = 3s_i^2 \, ds \] where \( ds = 0.5 \). Thus, the differential change in volume becomes \[ dV = 3s_i^2 \, \times 0.5 \].

Compute Metal Volume Using Differential Approximation

First, find \( s_i = \sqrt[3]{40}\) and square it to find \( s_i^2 \). Plug the result into the expression for \( dV \) to find the approximate volume of metal:\[ dV = 3(\sqrt[3]{40})^2 \, \times 0.5 \]. Calculate this using a calculator for an approximate numerical result.

Key concepts

Volume of a Cube

A cube is a three-dimensional shape with all sides of equal length. This makes calculating its volume straightforward; you simply multiply the length of one side by itself twice:

• The formula is: \( V = s^3 \)
• Here, \( V \) is the volume and \( s \) is the side length.

This formula represents how much space the cube occupies. For example, if the side of a cube measures 2 inches, then its volume is \( 2^3 = 8 \) cubic inches. The same principle applies to any cube, regardless of how small or large it is.
Understanding this simple equation is crucial, particularly when dealing with objects like boxes or containers, where volume dictates capacity.

Cube Root

Finding the cube root is the reverse process of cubing a number. It means finding a number which, when multiplied by itself three times, gives the original number.

• For instance, the cube root of 8 is 2 because \( 2^3 = 8 \).
• Mathematically, it is represented as \( \sqrt[3]{x} \).

In problems involving volume, such as determining the dimensions of a cubical box, identifying the cube root helps find the side length when the cube's volume is known.
Consider the problem we have: the interior volume is 40 cubic inches. To figure out the length of one side of the box (excluding thickness), you compute \( \sqrt[3]{40} \). This step is critical to find the precise measurement needed when designing or analyzing cubes in practical situations.

Thickness

Thickness in the context of a cube refers to the extent of material added to each side. For a cubical object consisting of a core and a covering layer (like metal), this thickness broadens each side.

• It contributes to both the total dimensions and the additional material volume.
• In the given problem, each wall of the box is 0.25 inch thick.

Upon computing the interior side length using the cube root, we determine the exterior side length by adding the total thickness.
Thus, if your interior side is 3 inches, and thickness is 0.25 inches per side, for two sides, you add 0.5 inches. This approach ensures the final dimension accounts for all materials used, making it essential in real-world manufacturing and construction scenarios.

Metal Volume Calculation

To determine the amount of metal used in creating the box, we must calculate the volume of the material alone. This involves a neat interplay of geometry and calculus using differentials.

• Start by finding the cubic volume of both the interior and exterior.
• The difference represents the metal volume.
• For small changes, differentials provide an approximation: \( dV = 3s_i^2 \cdot ds \).

Here, \( s_i \) is the interior side length and \( ds \) represents the additional volume introduced by thickness.
This approximation simplifies calculations, especially regarding slight variations in dimensions. Plug in your values for \( s_i \) and \( ds \) to find that \( dV \), which approximates how much metal encases the hollow interior.