Problem 35 - Intermediate Algebra

Question

Find the volume of each figure to the nearest hundredth. See Example 4. (Answers may vary slightly depending on which approximation of is used.) A rectangular solid with dimensions \(2.51 \mathrm{ft}, 3.71 \mathrm{ft},\) and \(10.21 \mathrm{ft}\)

Step-by-Step Solution

Answer

The volume of the rectangular solid is approximately 95.05 cubic feet.

Step 1: Understand the Formula for Volume of a Rectangular Solid

The volume of a rectangular solid can be found using the formula \( V = l \times w \times h \), where \( l \) is the length, \( w \) is the width, and \( h \) is the height of the solid.

Step 2: Substitute the Given Values into the Formula

Given dimensions are \( l = 2.51 \text{ ft} \), \( w = 3.71 \text{ ft} \), and \( h = 10.21 \text{ ft} \). Substitute these values into the volume formula: \( V = 2.51 \times 3.71 \times 10.21 \).

Step 3: Perform the Multiplication

First, calculate \( 2.51 \times 3.71 \approx 9.3121 \). Next, multiply the result by 10.21: \( 9.3121 \times 10.21 \approx 95.054841 \).

Step 4: Round to the Nearest Hundredth

The calculated volume \( 95.054841 \) should be rounded to the nearest hundredth (two decimal places), resulting in \( 95.05 \text{ ft}^3 \).

Key Concepts

Understanding Rectangular Solids

A rectangular solid is a 3-dimensional shape with 6 faces, all of which are rectangles. It's also known as a box or a cuboid.
These shapes have three key dimensions:

• Length (l)
• Width (w)
• Height (h)

Each pair of opposite faces are identical rectangles, and all angles are right angles.
This regularity makes it straightforward to perform calculations related to it, like finding its volume.

Exploring the Volume Formula

The formula for finding the volume of a rectangular solid is crucial for accurately determining how much space it occupies.
The volume formula:\[ V = l \times w \times h \]Where:

• \( l \) = length
• \( w \) = width
• \( h \) = height

This formula essentially multiplies the three dimensions together to get the total volume. The volume gives us a measure of how much space is contained within or how much material can fit inside the object. This calculation helps in fields ranging from packing and shipping to construction and manufacturing.

Importance of Mathematical Rounding

Rounding numbers is an essential step in mathematics, especially when dealing with measurements that require precision.
When asked to round to the nearest hundredth, we'll look at the third decimal place:

• If it is 5 or greater, we round up
• If it is less than 5, we round down

For instance, in the volume calculation outcome of 95.054841, rounding to the nearest hundredth results in 95.05. This step ensures that the numbers are easier to understand and use, especially in practical contexts where exactness is less critical than clarity.

The Multiplication Process in Volume Calculation

In calculating the volume, multiplication is the key operation. Each step involves multiplying two numbers:

• First, multiply the length and width
• Second, take that product and multiply by the height

For our example: - Multiply 2.51 by 3.71 to get approximately 9.3121. - Then, multiply 9.3121 by 10.21 to achieve roughly 95.054841.
This sequential multiplication ensures that you consider the contribution of each dimension fully. Understanding each step in the process can help avoid common mistakes, like missing a dimension or using incorrect arithmetic operations.