Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
Find the surface area of the triangular prism. Be sure to include the correct unit in the answer.
![### Calculating the Surface Area of a Triangular Prism
To determine the surface area of this triangular prism, we need to consider both the lateral and the base areas.
#### Dimensions Provided:
- Length of the triangular base (adjacent side): 9 yards
- Height of the triangle base (perpendicular to the 9-yard side): 6 yards
- Hypotenuse of the triangular base: 10 yards
- Height (or length) of the prism: 8 yards
#### Detailed Explanation of the Diagram:
The illustration provided depicts a triangular prism. This prism has a triangular base with the following sides:
- One side measures 9 yards,
- Another side (height of the triangle) is 6 yards,
- The hypotenuse is 10 yards.
The prism itself extends to a height (or length) of 8 yards.
#### Steps to Calculate Surface Area:
To find the surface area of a triangular prism, we need to calculate the areas of the two triangular bases and the three rectangular sides, then sum them up.
1. **Calculate the Area of the Triangular Base:**
The area \( A \) of a triangle is given by the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
Substitute the values:
\[
A = \frac{1}{2} \times 9 \text{ yd} \times 6 \text{ yd} = 27 \text{ yd}^2
\]
Since there are two triangular bases, the total area of both triangular bases is:
\[
2 \times 27 \text{ yd}^2 = 54 \text{ yd}^2
\]
2. **Calculate the Lateral Surface Area:**
The lateral surface area is the sum of the areas of the three rectangles formed by the prism’s height and the sides of the triangular base (9 yards, 6 yards, and 10 yards).
\[
\text{Area of rectangle with side 9 yd} = 9 \text{ yd} \times 8 \text{ yd} = 72 \text{ yd}^2
\]
\[
\text{Area of rectangle with side 6 yd} = 6 \text{ yd} \times 8 \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7745b586-061e-4aaa-b5fb-1a2455772746%2F94591033-b1f7-48c7-ad22-0ac9238d5960%2Fmz8fgv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculating the Surface Area of a Triangular Prism
To determine the surface area of this triangular prism, we need to consider both the lateral and the base areas.
#### Dimensions Provided:
- Length of the triangular base (adjacent side): 9 yards
- Height of the triangle base (perpendicular to the 9-yard side): 6 yards
- Hypotenuse of the triangular base: 10 yards
- Height (or length) of the prism: 8 yards
#### Detailed Explanation of the Diagram:
The illustration provided depicts a triangular prism. This prism has a triangular base with the following sides:
- One side measures 9 yards,
- Another side (height of the triangle) is 6 yards,
- The hypotenuse is 10 yards.
The prism itself extends to a height (or length) of 8 yards.
#### Steps to Calculate Surface Area:
To find the surface area of a triangular prism, we need to calculate the areas of the two triangular bases and the three rectangular sides, then sum them up.
1. **Calculate the Area of the Triangular Base:**
The area \( A \) of a triangle is given by the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
Substitute the values:
\[
A = \frac{1}{2} \times 9 \text{ yd} \times 6 \text{ yd} = 27 \text{ yd}^2
\]
Since there are two triangular bases, the total area of both triangular bases is:
\[
2 \times 27 \text{ yd}^2 = 54 \text{ yd}^2
\]
2. **Calculate the Lateral Surface Area:**
The lateral surface area is the sum of the areas of the three rectangles formed by the prism’s height and the sides of the triangular base (9 yards, 6 yards, and 10 yards).
\[
\text{Area of rectangle with side 9 yd} = 9 \text{ yd} \times 8 \text{ yd} = 72 \text{ yd}^2
\]
\[
\text{Area of rectangle with side 6 yd} = 6 \text{ yd} \times 8 \
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