Ch. R - Review of Basic Concepts

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. R - Review of Basic Concepts

Problem 117

Chapter 1, Problem 117

Answer each question. If the lengths of the sides of a cube are tripled, by what factor will the volume change?

Verified step by step guidance

Recall the formula for the volume of a cube: $V = s^3$, where $s$ is the length of a side of the cube.

If the side length is tripled, the new side length becomes \$3s$.

Substitute the new side length into the volume formula to find the new volume: $V_{new} = (3s)^3$.

Simplify the expression for the new volume: $V_{new} = 3^3 \times s^3$.

Since $3^3 = 27$, the volume changes by a factor of 27 compared to the original volume.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Volume of a Cube

The volume of a cube is calculated by cubing the length of one of its sides (V = s³). This formula shows how volume depends on the side length, making it essential to understand how changes in side length affect volume.

Scaling Factor

A scaling factor describes how dimensions of a shape change when scaled up or down. When the side length of a cube is multiplied by a factor, the volume changes by that factor raised to the third power, since volume is three-dimensional.

Effect of Dimension Changes on Volume

When the side length of a cube is tripled, the volume increases by the cube of 3, which is 27. This illustrates how volume grows exponentially with linear dimension changes, emphasizing the cubic relationship between side length and volume.

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