Problem 31 - Calculus

Question

The cube \(0 \leqslant x, y, z \leqslant s\) has volume \(V=\) ______. The three square faces with \(x=s\) or \(y=s\) or \(z=s\) have total area \(A=\) _______. If \(s\) is increased by \(\Delta s\), the extra volume has the shape of ___________. That volume \(\Delta V\) is approximately _______ .So \(d V / d s=\) _______.

Step-by-Step Solution

Answer

\(V = s^3\), \(A = 3s^2\), cubic shape, \(\Delta V \approx 3s^2 \Delta s\), \(dV/ds = 3s^2\).

Step 1: Calculate the Volume of the Cube

The volume of a cube is proportional to the cube of the side length. Here the cube has side length \( s \), so its volume \( V \) is calculated as follows:\[V = s^3\]

Step 2: Calculate the Area of the Three Square Faces

The squares with sides on the planes \( x = s \), \( y = s \), and \( z = s \) each have side lengths \( s \) and area \( s^2 \). Since there are 3 such squares, the total area \( A \) of these faces is given by:\[A = 3s^2\]

Step 3: Describe the Shape of the Extra Volume

When \( s \) is increased by \( \Delta s \), additional volume is added around the cube. This extra volume forms a shape of a cuboid (rectangular prism) that wraps around the original cube with three distinct sections: one along each of the three planes where the original faces were.

Step 4: Calculate the Extra Volume

The extra volume consists of three distinct parts:1. Three rectangular prisms along each face increment: each with area \( s \times \Delta s \) and height \( s \), totaling 3 such volumes.2. Three edges where two faces meet, each with area \( \Delta s \times \Delta s \) and extending the length of the side \( s \), totaling 3 such volumes.3. A small cube in the corner added by the extra \( \Delta s \) on all three axes, with volume \( (\Delta s)^3 \).Thus,\[\Delta V = 3(s^2 \Delta s) + 3(s \Delta s^2) + (\Delta s)^3 \]

Step 5: Approximate the Change in Volume for Small \\Delta s

For small \( \Delta s \), the higher-order term \( (\Delta s)^3 \) becomes negligible. Therefore, we approximate:\[\Delta V \approx 3s^2 \Delta s\]

Step 6: Find the Derivative of the Volume with Respect to \\s

The derivative of the volume with respect to \( s \) measures how the volume changes as \( s \) changes. The derivative \( \frac{dV}{ds} \) is derived from the expression for \( \Delta V \):\[\frac{dV}{ds} = 3s^2\]

Key Concepts

Volume of a Cube

The volume of a cube is a fundamental concept in geometry. It refers to the amount of space enclosed within a cube. A cube is a special type of 3-dimensional shape where each of the six faces is a square and all edges are of equal length. To find the volume of a cube, use the formula:\[ V = s^3 \] where \( s \) is the length of one side of the cube. This formula tells us that the volume is equal to the length of a side multiplied by itself twice. For example, if each side of a cube measures 2 units, the volume is calculated as \( 2^3 = 8 \) cubic units. Understanding this basic principle allows you to extend the idea of volume to other shapes.

Surface Area

The surface area of a cube can be visualized as the total area covered by its six faces, which are all equal squares. In the specific exercise problem, we focus on only three of these square faces, specifically those positioned on the planes \( x = s \), \( y = s \), and \( z = s \). Each of these faces has an area calculated by squaring the side length \( s \), so:\[ A_{ ext{face}} = s^2 \] Since there are three such faces, the total area \( A \) for these specific squares is:\[ A = 3s^2 \] Surface area plays an important role in understanding how to enclose or cover a 3-dimensional object. It is essential for solving problems related to painting, wrapping, or even airflow around objects.

Rectangular Prism

When dealing with changes to a cube, it is practical to consider the additional volume created when a cube's side length is increased by a small increment \( \Delta s \). This new volume forms a geometric shape called a rectangular prism or cuboid. This shape can be thought of as wrapping around the existing cube and consists of three types of sections:

• Three rectangular faces each facing one of the cube's axes.
• Three prisms along the edges where two faces meet.
• A small cube at the corner formed by the extension along three axes.

Understanding the formation of this new shape helps in calculating the extra volume precisely and is an effective way to visualize the space changes in a cube.

Approximate Calculation

In mathematical problems, especially those involving derivatives, simplifying complex expressions is a key strategy. When the side length \( s \) of a cube is increased by a tiny amount \( \Delta s \), calculating the exact extra volume can be difficult. However, for small changes, higher-order terms like \( (\Delta s)^3 \) can be ignored. Thus, the extra volume, \( \Delta V \), becomes:\[ \Delta V \approx 3s^2 \Delta s \] This approximation simplifies the calculation significantly, allowing for easier comparison or extension into calculus concepts like differentiation. By focusing on substantial terms, you achieve an efficient way to analyze small changes without unnecessary complexity.