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Chapter 8: Problem 19

Determine the volume of each right prism described. Regular hexagon base with side 3.6 yd; height 4.8 yd

Short Answer

Expert verified
The volume is approximately 161.57 cubic yards.

Step by step solution

01

- Determine the Area of the Hexagonal Base

A regular hexagon can be divided into 6 equilateral triangles. The area of one equilateral triangle with side length 3.6 yards is calculated using the formula: \[ \text{Area} = \frac{\sqrt{3}}{4} s^2 \] where \( s = 3.6 \) yards. Calculate this area first: \[ \text{Area of one triangle} = \frac{\sqrt{3}}{4} (3.6)^2 \] \[ = \frac{\sqrt{3}}{4} \times 12.96 \] \[ = 3.24\sqrt{3} \text{ square yards} \] Multiply by 6 to get the total area of the hexagon: \[ \text{Area of hexagon} = 6 \times 3.24\sqrt{3} \] \[ = 19.44\sqrt{3} \text{ square yards} \]
02

- Calculate the Volume of the Prism

The volume of a prism is found by multiplying the area of the base by the height. We already calculated the area of the hexagonal base to be \( 19.44\sqrt{3} \) square yards. The height of the prism is given as 4.8 yards. Use the following formula to find the volume: \[ \text{Volume} = \text{Base Area} \times \text{Height} \] \[ \text{Volume} = 19.44\sqrt{3} \times 4.8 \] Calculate the result: \[ \text{Volume} = 93.312\sqrt{3} \text{ cubic yards} \] \[ \approx 161.57 \text{ cubic yards} \]

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Key Concepts

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

Hexagonal Base
A hexagonal base is a six-sided polygon where all sides and angles are equal, known as a regular hexagon.
To break down a hexagon for easier calculation, think of it as being composed of 6 equilateral triangles.
Each of these triangles has the same side length, making it easier to determine the area.
This side length is crucial in further calculations. In the given problem, the side length is 3.6 yards.
Visualizing this division can aid in understanding how the total area of the hexagon is derived.
Area Calculation
To find the area of a regular hexagon, we first find the area of one of its equilateral triangles.
The formula for the area of an equilateral triangle is: \( \text{Area} = \frac{\sqrt{3}}{4} s^2 \)
where \( s \) is the side length. For our problem, \( s = 3.6 \) yards.
Substituting the value, we get:
\[ \text{Area of one triangle} = \frac{\sqrt{3}}{4} (3.6)^2 = 3.24\sqrt{3} \text{ square yards} \]
Since the hexagon is composed of 6 such triangles, we multiply the area of one triangle by 6:
\[ \text{Area of hexagon} = 6 \times 3.24\sqrt{3} = 19.44\sqrt{3} \text{ square yards} \]
This is the total area of the hexagonal base.
Volume Formula
When calculating the volume of a right prism, we use the formula: \( \text{Volume} = \text{Base Area} \times \text{Height} \)
The height of the prism is the perpendicular distance between its bases, given as 4.8 yards.
Using the previously calculated base area \( 19.44\sqrt{3} \text{ square yards} \), we find the volume as:
\[ \text{Volume} = 19.44\sqrt{3} \times 4.8 = 93.312\sqrt{3} \text{ cubic yards} \]
Approximating the value, we get the final volume: \[ \text{Volume} \approx 161.57 \text{ cubic yards} \]
Thus, the volume of the right prism with a hexagonal base is approximately 161.57 cubic yards.

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