Problem 50 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

Estimate what will happen to the volume of a cube if the length of each side is decreased by \(2 \%\). Express your answer as a percentage.

Step-by-Step Solution

Answer

The volume decreases by approximately 5.9%.

Step 1: Understand the volume formula for a cube

The volume of a cube is calculated by the formula: \[ V = a^3 \] where \(a\) is the length of one side of the cube.

Step 2: Express the new side length

If the side length is decreased by 2%, the new side length will be: \[ a_{\text{new}} = a \times (1 - 0.02) = 0.98a \]

Step 3: Write the volume formula with the new side length

Using the new side length, the new volume \(V_{\text{new}}\) is: \[ V_{\text{new}} = (0.98a)^3 \]

Step 4: Simplify the new volume formula

Simplify the expression to find the new volume: \[ V_{\text{new}} = 0.98^3 \times a^3 \]

Step 5: Calculate the change in volume

Calculate the value of \(0.98^3\): \[ 0.98^3 \ = 0.98 \times 0.98 \times 0.98 \ ≈ 0.941 \] Therefore, \[ V_{\text{new}} ≈ 0.941 \times a^3 \]

Step 6: Determine the percentage change in volume

The original volume is \(V = a^3\), and the new volume is approximately \(0.941 a^3\). To find the percentage change: \[ \text{Percentage change} = \frac{V - V_{\text{new}}}{V} \times 100\ = \frac{a^3 - 0.941a^3}{a^3} \times 100 \ = (1 - 0.941) \times 100 \ = 0.059 \times 100 \ = 5.9\text{%} \]

Key Concepts

concept 1: volume of a cube

Understanding the volume of a cube is fundamental in this exercise. A cube is a special type of three-dimensional shape where all sides are equal in length. The volume of a cube represents the amount of space occupied within the cube. It can be calculated using the formula: \[ V = a^3 \]where \(V\) is the volume and \(a\) is the length of one side of the cube. The exponent 3 signifies that all three dimensions (length, width, and height) are the same. This formula is straightforward and tells us how much space is inside the cube based on the side length.

concept 2: percentage decrease

A percentage decrease helps us understand how much something has reduced by a certain percentage. In this exercise, the length of each side of the cube is decreased by \(2\%\). To find the new side length, we use the formula: \[ a_{\text{new}} = a \times (1 - 0.02) = 0.98a \]This equation means that each side of the cube is now reduced to \(98\%\) of its original length. The factor \(0.98\) represents the remaining length after a \(2\%\) decrease. This reduced side length will then be used to calculate the new volume of the cube.

concept 3: derivatives and approximations

Derivatives and approximations give us tools to understand changes in mathematical functions. Here, we don't directly use derivatives, but understanding how changes in one quantity affect another is at the core of calculus.When we decrease the side length by \(2\%\), we see how it affects the entire volume of the cube. By plugging the new side length into the volume formula, we get the new volume: \[ V_{\text{new}} = (0.98a)^3 \]Simplifying, it becomes: \[ V_{\text{new}} = 0.98^3 \times a^3 \]Calculating \(0.98^3\) gives us approximately \(0.941\). Therefore, the new volume is roughly \(0.941 \times a^3\), showing a decrease in volume. The percentage change can be calculated as: \[ \text{Percentage change} = \frac{V - V_{\text{new}}}{V} \times 100 = (1 - 0.941) \times 100 \text{ which equals } 5.9\text{\text{%}} \]This shows that a \(2\%\) decrease in side length results in approximately a \(5.9\%\) decrease in the volume.