Problem 78 - Calculus Volume 1

Question

Find the change in volume \(d V\) or in surface area \(d A\). \(d V\) if the sides of a cube change from 10 to 10.1 .

Step-by-Step Solution

Answer

The change in volume, \(d V\), is 30.301 cubic units.

Step 1: Understand the Problem

We have a cube whose side length changes from 10 units to 10.1 units. We need to find the change in volume, denoted by \(d V\).

Step 2: Calculate Initial Volume

The volume \(V\) of a cube with side length \(s\) is given by the formula \(V = s^3\). For a cube with side length 10, it is \(V_1 = 10^3 = 1000\).

Step 3: Calculate Final Volume

After the change in side length, the new side length is 10.1. The final volume \(V_2\) is \(10.1^3\). Calculate this value: \(V_2 = 10.1 \times 10.1 \times 10.1 = 1030.301\).

Step 4: Find the Change in Volume

The change in volume \(d V\) is the difference between the final volume and the initial volume. Hence, \(d V = V_2 - V_1 = 1030.301 - 1000\). Simplify the expression to get \(d V = 30.301\).

Key Concepts

Cube Volume Calculation

Calculating the volume of a cube is an essential part of understanding how volume changes when dimensions change. A cube is a three-dimensional shape with all sides equal. This equality in dimensions simplifies our calculations. You can calculate a cube's volume using the formula: \[ V = s^3 \]- **V** represents the volume of the cube.- **s** is the length of a side of the cube.By raising the side length to the power of three, we effectively calculate how much three-dimensional space the cube occupies. For example, a cube with a side length of 10 units has a volume of \(10^3 = 1000\) cubic units. Understanding this basic calculation is crucial before moving to more complex concepts like how volume changes with differentials.

Volume Change

Volume change in a cube occurs when the cube's side length changes. It's a straightforward computation once you have the initial and final volumes. When a cube changes its side from one length to another, the volume change can be denoted as \(dV\), symbolizing the difference in volume due to a tiny change in the side length. To compute the volume change:- Calculate the initial volume \(V_1\) with the original side length.- Calculate the final volume \(V_2\) after the side length has changed.The change in volume can then be found using the formula:\[ dV = V_2 - V_1 \]For a cube whose side changes from 10 to 10.1, as mentioned:- Initially, \(V_1 = 1000 \)- Finally, \(V_2 = 1030.301\)Thus, \(dV = 1030.301 - 1000 = 30.301\). Understanding volume change highlights the impact of even small adjustments in measurements.

Differentiation Techniques

Differentiation is a vital technique in calculus used to determine how a function changes as its inputs change. In the context of volume calculation, differentiation can help us understand how the volume of a cube changes with a small change in side length. The key to differentiation is finding the derivative of a function. For a cube, the volume function is:\[ V = s^3 \]When we differentiate this with respect to \(s\), we find:\[ \frac{dV}{ds} = 3s^2 \]This derivative tells us the rate of change of volume (\(dV\)) with respect to the change in side (\(ds\)). This concept is pivotal for identifying how tiny changes affect overall measures. Even in everyday scenarios, understanding differentiation techniques enables precise predictions and enhancements in design, manufacturing, and other fields.