Question
Question
Use the image to answer the question.
An illustration shows a triangular prism with length 6 inches, width 2 inches, perpendicular height of the triangular face 1.12 inches, and side of triangular face 1.5 inches.
What is the total surface area of this eraser? Your answer should have two decimal places.
(1 point)
in.2
Incorrect
First, find the area of one triangle using the formula A=12 b×h, then double the result since there are two identical bases.
Use the image to answer the question.
An illustration shows a triangular prism with length 6 inches, width 2 inches, perpendicular height of the triangular face 1.12 inches, and side of triangular face 1.5 inches.
What is the total surface area of this eraser? Your answer should have two decimal places.
(1 point)
in.2
Incorrect
First, find the area of one triangle using the formula A=12 b×h, then double the result since there are two identical bases.
Answers
Answered by
GPT-4o mini
To find the total surface area of the triangular prism, we will calculate the areas of the triangular bases and the rectangular lateral faces.
1. **Area of the triangular bases**:
The area \( A \) of one triangle can be calculated using the formula:
\[
A = \frac{1}{2} \times b \times h
\]
where \( b \) is the base of the triangle and \( h \) is the height.
Given:
- Base \( b = 1.5 \) inches
- Height \( h = 1.12 \) inches
Substituting the values:
\[
A = \frac{1}{2} \times 1.5 \times 1.12 = \frac{1.68}{2} = 0.84 \text{ square inches}
\]
Since there are two triangular bases:
\[
\text{Total area of triangular bases} = 2 \times 0.84 = 1.68 \text{ square inches}
\]
2. **Area of the rectangular lateral faces**:
There are three rectangular faces in the prism, corresponding to the three sides of the triangular base.
- For the rectangular face with height equal to the length of the prism:
- Base \( = 1.5 \) inches
- Area = Length × Height = \( 6 \times 1.5 = 9.0 \) square inches
- For the second rectangular face with height equal to the length of the prism:
- Since we need the height corresponding to each side of the triangle, we have:
1. The side \( = 1.5 \) inches (same as above)
2. The other two sides of the triangular face (the width of the prism \( = 2 \))
- The other rectangular faces are:
- Height = \( 2 \) inches:
1. Area = Length × Height = \( 6 \times 2 = 12.0 \) square inches
Continuing with all sides of the triangle:
- We must calculate areas for each:
- Side 1: \( 6 \times 1.5 = 9.0 \)
- Side 2: \( 6 \times 2 = 12.0 \)
- Side 3 (the remaining is equal to the hypotenuse, finding through Pythagoras): Calculate the hypotenuse first (Pythagorean Theorem):
- \( c = \sqrt{(1.5)^2 + (1.12)^2} = \sqrt{2.25 + 1.2544} = \sqrt{3.5044} \approx 1.87 \)
- Hence the area is \( 6 \times 1.87 = 11.22 \)
Now summing the areas of the rectangular faces:
\[
\text{Total area of rectangular faces} = 9 + 12 + 11.22 = 32.22
\]
3. **Total Surface Area of the Prism**:
Adding the areas of the triangular bases and the rectangular lateral faces:
\[
\text{Total Surface Area} = \text{Area of triangular bases} + \text{Area of lateral faces} = 1.68 + 32.22 = 33.90 \text{ square inches}
\]
Thus, the total surface area of the prism is:
\[
\boxed{33.90}
\]
1. **Area of the triangular bases**:
The area \( A \) of one triangle can be calculated using the formula:
\[
A = \frac{1}{2} \times b \times h
\]
where \( b \) is the base of the triangle and \( h \) is the height.
Given:
- Base \( b = 1.5 \) inches
- Height \( h = 1.12 \) inches
Substituting the values:
\[
A = \frac{1}{2} \times 1.5 \times 1.12 = \frac{1.68}{2} = 0.84 \text{ square inches}
\]
Since there are two triangular bases:
\[
\text{Total area of triangular bases} = 2 \times 0.84 = 1.68 \text{ square inches}
\]
2. **Area of the rectangular lateral faces**:
There are three rectangular faces in the prism, corresponding to the three sides of the triangular base.
- For the rectangular face with height equal to the length of the prism:
- Base \( = 1.5 \) inches
- Area = Length × Height = \( 6 \times 1.5 = 9.0 \) square inches
- For the second rectangular face with height equal to the length of the prism:
- Since we need the height corresponding to each side of the triangle, we have:
1. The side \( = 1.5 \) inches (same as above)
2. The other two sides of the triangular face (the width of the prism \( = 2 \))
- The other rectangular faces are:
- Height = \( 2 \) inches:
1. Area = Length × Height = \( 6 \times 2 = 12.0 \) square inches
Continuing with all sides of the triangle:
- We must calculate areas for each:
- Side 1: \( 6 \times 1.5 = 9.0 \)
- Side 2: \( 6 \times 2 = 12.0 \)
- Side 3 (the remaining is equal to the hypotenuse, finding through Pythagoras): Calculate the hypotenuse first (Pythagorean Theorem):
- \( c = \sqrt{(1.5)^2 + (1.12)^2} = \sqrt{2.25 + 1.2544} = \sqrt{3.5044} \approx 1.87 \)
- Hence the area is \( 6 \times 1.87 = 11.22 \)
Now summing the areas of the rectangular faces:
\[
\text{Total area of rectangular faces} = 9 + 12 + 11.22 = 32.22
\]
3. **Total Surface Area of the Prism**:
Adding the areas of the triangular bases and the rectangular lateral faces:
\[
\text{Total Surface Area} = \text{Area of triangular bases} + \text{Area of lateral faces} = 1.68 + 32.22 = 33.90 \text{ square inches}
\]
Thus, the total surface area of the prism is:
\[
\boxed{33.90}
\]