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Chapter 9: Problem 15

Find the volume of a cube whose side is 3 yards in length. (a) \(9 \mathrm{yd}^{3}\) (b) \(12 \mathrm{yd}^{3}\) (c) \(18 \mathrm{yd}^{3}\) (d) \(27 \mathrm{yd}^{3}\)

Short Answer

Expert verified
The volume of the cube is 27 yd鲁, which is option (d).

Step by step solution

01

Understand the Cube

A cube is a three-dimensional shape with all sides equal in length. To find the volume of a cube, you multiply the length of one side by itself twice (since it's in three dimensions).
02

Apply the Volume Formula

The formula for the volume of a cube is given by \( V = s^3 \), where \( s \) is the length of one side of the cube. Here, \( s = 3 \) yards.
03

Perform the Calculation

Substitute the value of \( s \) into the formula: \( V = (3)^3 = 3 \times 3 \times 3 \).
04

Compute the Result

Calculate the cube of 3: \( 3 \times 3 = 9 \), and then \( 9 \times 3 = 27 \). So, the volume \( V = 27 \) cubic yards.
05

Choose the Correct Option

The computed volume is \( 27 \text{ yd}^3 \), which corresponds to option (d).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Three-Dimensional Shapes
Three-dimensional shapes, also known as 3D shapes, are objects that have depth, width, and height. Unlike 2D shapes that have only a length and a width, 3D shapes occupy space in the physical world. A cube is a perfect example of a three-dimensional shape.
Each side of a cube is a square, meaning all sides are equal in length.
  • This makes the cube unique, as it has the same measurement in each direction: length, width, and height.
  • The three-dimensional nature of a cube means it can hold or contain things, much like a box does.
When you look at a cube, you can see its ability to hold a certain volume, which we can calculate with a specific formula.
The Volume Formula for a Cube
Finding the volume of any three-dimensional object helps us understand how much space it occupies.
For a cube, calculating the volume is straightforward due to its uniform dimensions. The core formula to determine the volume of a cube is:
  • \[ V = s^3 \]
where \( V \) represents the volume, and \( s \) is the length of one of the cube's sides.
Because a cube's sides are all identical, we multiply the side length by itself three times:
  • First, square the side length to find the area of one face.
  • Then, multiply that area by the side length again, accounting for the depth of the cube.
This formula gives us a direct way to calculate how much space the cube takes up in three dimensions.
Practical Mathematics Problem Solving
Solving mathematics problems involves a simple step-by-step approach, ensuring clear understanding and accurate calculations.
Let's break down the solution to finding a cube's volume.
First, identify what the question is asking: in our exercise, it asks for the cube's volume given a side length.
  • Identify known values (the side of the cube is 3 yards).
  • Apply the appropriate formula ( \[ V = s^3 \] ) using these values.
Next, perform the calculations:
  • Compute \( 3 imes 3 \ = 9 \) and then the final step, \( 9 imes 3 = 27 \) .
All mathematical problem-solving involves checking and verifying answers.
Finally, confirm your answer matches one of the provided options, ensuring that your understanding and calculations are correct.
This step-by-step problem-solving process builds a foundation for tackling even more complex mathematics challenges.

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Most popular questions from this chapter

Find the volume of a cylinder whose base has a radius of 4 inches and whose height is 9 inches. Use \(\pi=3.14\). (a) 452.16 in. \(^{3}\) (b) 226.08 in. (c) 113.04 in. \(^{3}\) (d) 1017.36 in. \(^{3}\)

Find the circumference of a circle whose diameter is 24 inches. Use \(\pi=3.14 .\) (a) 37.68 in. (b) 75.36 in. (c) 452.16 in. (d) 1808.64 in.

Find the area of a triangle whose base is 42 yards and whose height is 36 yards. (a) \(78 \mathrm{yd}^{2}\) (b) \(39 \mathrm{yd}^{2}\) $$ \text { (c) } 756 \mathrm{yd}^{2} $$ (d) \(1512 \mathrm{yd}^{2}\)

How many square inches are there in 15 square feet? (a) 2160 in. \(^{2}\) (b) 180 in. \(^{2}\) (c) 225 in. \(^{2}\) (d) 135 in. \(^{2}\)

Find the area of a square whose side is 6.2 feet. (a) \(38.44 \mathrm{ft}^{2}\) (b) \(24.8 \mathrm{ft}^{2}\) (c) \(1.55 \mathrm{ft}^{2}\) (d) \(12.4 \mathrm{ft}^{2}\)
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