Question

A brass rectangular solid measures \(4.95 \mathrm{~cm}\) by \(2.45 \mathrm{~cm} .\) What is the thickness of the brass solid if the volume is \(10.0 \mathrm{~cm}^{3} ?\)

Short answer

The thickness of the brass solid is approximately \(0.824 \, \mathrm{cm}\).

Step-by-step solution

Understanding the Formula

To find the thickness of a rectangular solid given its volume, length, and width, you need to use the formula for volume of a rectangular solid, which is: \[V = ext{Length} \times ext{Width} \times ext{Thickness}\]

Substituting Known Values

We know the volume \(V\) is \(10.0 \, \mathrm{cm}^3\), the length is \(4.95 \, \mathrm{cm}\), and the width is \(2.45 \, \mathrm{cm}\). Substitute these into the volume formula:\[10.0 = 4.95 \times 2.45 \times \text{Thickness}\]

Isolating Thickness

To solve for the thickness, rearrange the equation:\[\text{Thickness} = \frac{10.0}{4.95 \times 2.45}\]

Calculating

Calculate the denominator \(4.95 \times 2.45\) and then divide the volume by this product:\[4.95 \times 2.45 = 12.1275 \\text{Thickness} = \frac{10.0}{12.1275} \\text{Thickness} \approx 0.824 \, \mathrm{cm}\]

Key concepts

Understanding Rectangular Solids

A rectangular solid, also known as a cuboid, is a three-dimensional shape that has six faces, all of which are rectangles. It resembles a box-like structure with specific measurements for length, width, and height (or thickness).
The characteristics of a rectangular solid are:

• It has three dimensions: length, width, and thickness (or height).
• All faces are rectangles, and opposite faces are equal.
• The volume of a rectangular solid is calculated by multiplying its length, width, and thickness.

This geometric shape is prevalent in our daily lives and can be found in objects like bricks, books, and boxes. Understanding its structure and properties is crucial for tasks involving volume and area calculations.

Calculating Thickness of a Rectangular Solid

Thickness refers to the measure of the solid object from top to bottom, in the direction perpendicular to the length and width. When given the volume of a rectangular solid along with its length and width, we can find the thickness using the volume formula.
In our problem, the volume is provided, and so are the other two dimensions. The formula to determine thickness is:

• Volume = Length × Width × Thickness

This can be rearranged to solve for thickness:

• Thickness = \( \frac{\text{Volume}}{\text{Length} \times \text{Width}} \)

Applying the provided numbers:

• Substitute Volume = 10.0 cm³, Length = 4.95 cm, Width = 2.45 cm into the formula.
• Calculate Thickness: \( \frac{10.0}{4.95 \times 2.45} \approx 0.824 \text{ cm} \)

This thickness determination process is a common exercise in geometry where solving for unknown dimensions of solid objects is required.

Using Geometry Formulas for Problem Solving

Geometry formulas are powerful tools that help us solve problems involving different shapes and structures. For rectangular solids, knowing the formulas for calculating volume, surface area, and other dimensions can simplify seemingly complex problems.
The volume formula for a rectangular solid, \( V = \text{Length} \times \text{Width} \times \text{Thickness} \), serves multiple purposes beyond just finding volume. It enables calculation of any missing dimension when the other measurements are known. Here are a few practical applications:

• Determining the amount of material needed to fill a space.
• Calculating storage capacity or fluid volume an object can hold.
• Finding missing dimensions when remodeling or designing structures.

By understanding how to rearrange and apply these formulas, you can tackle a wide range of geometric problems related to solid objects efficiently. Mastering these concepts can significantly enhance problem-solving skills, especially in engineering, design, and architecture fields.