Triangular Prisms (A) - Calculate volume and surface area using given formulas.

To solve the problem of calculating the volume and surface area of each triangular prism, we will follow these steps:

Step 1: Understand the Formulas

1. Volume of a Triangular Prism:
\[
V = \text{Area of Base} \times \text{Prism Length}
\]
The area of the triangular base is given by:
\[
\text{Area of Base} = 0.5 \times b \times h
\]
where \( b \) is the base of the triangle and \( h \) is the height of the triangle.

2. Surface Area of a Triangular Prism:
\[
\text{Surface Area} = (\text{Perimeter of Base} \times \text{Prism Length}) + (2 \times \text{Area of Base})
\]
- Perimeter of Base: Sum of all sides of the triangular base.
- Area of Base: As calculated above.

Step 2: Solve Each Problem

#### Problem 1:
- Dimensions:
- Base of triangle (\( b \)): 23.2 in
- Height of triangle (\( h \)): 40.1 in
- Prism Length (\( l \)): 28.6 in

1. Calculate the Area of the Base:
\[
\text{Area of Base} = 0.5 \times b \times h = 0.5 \times 23.2 \times 40.1 = 0.5 \times 931.32 = 465.66 \, \text{in}^2
\]

2. Calculate the Volume:
\[
V = \text{Area of Base} \times l = 465.66 \times 28.6 = 13347.636 \, \text{in}^3
\]

3. Calculate the Perimeter of the Base:
- Sides of the triangle: 23.2 in, 40.1 in, and 48.2 in
\[
\text{Perimeter of Base} = 23.2 + 40.1 + 48.2 = 111.5 \, \text{in}
\]

4. Calculate the Surface Area:
\[
\text{Surface Area} = (\text{Perimeter of Base} \times l) + (2 \times \text{Area of Base})
\]
\[
\text{Surface Area} = (111.5 \times 28.6) + (2 \times 465.66)
\]
\[
\text{Surface Area} = 3176.9 + 931.32 = 4108.22 \, \text{in}^2
\]

#### Problem 2:
- Dimensions:
- Base of triangle (\( b \)): 25.6 mm
- Height of triangle (\( h \)): 32.2 mm
- Prism Length (\( l \)): 19.7 mm

1. Calculate the Area of the Base:
\[
\text{Area of Base} = 0.5 \times b \times h = 0.5 \times 25.6 \times 32.2 = 0.5 \times 824.32 = 412.16 \, \text{mm}^2
\]

2. Calculate the Volume:
\[
V = \text{Area of Base} \times l = 412.16 \times 19.7 = 8120.752 \, \text{mm}^3
\]

3. Calculate the Perimeter of the Base:
- Sides of the triangle: 25.6 mm, 32.2 mm, and 39.0 mm
\[
\text{Perimeter of Base} = 25.6 + 32.2 + 39.0 = 96.8 \, \text{mm}
\]

4. Calculate the Surface Area:
\[
\text{Surface Area} = (\text{Perimeter of Base} \times l) + (2 \times \text{Area of Base})
\]
\[
\text{Surface Area} = (96.8 \times 19.7) + (2 \times 412.16)
\]
\[
\text{Surface Area} = 1906.96 + 824.32 = 2731.28 \, \text{mm}^2
\]

#### Problem 3:
- Dimensions:
- Base of triangle (\( b \)): 10.2 m
- Height of triangle (\( h \)): 11.0 m
- Prism Length (\( l \)): 19.8 m

1. Calculate the Area of the Base:
\[
\text{Area of Base} = 0.5 \times b \times h = 0.5 \times 10.2 \times 11.0 = 0.5 \times 112.2 = 56.1 \, \text{m}^2
\]

2. Calculate the Volume:
\[
V = \text{Area of Base} \times l = 56.1 \times 19.8 = 1110.18 \, \text{m}^3
\]

3. Calculate the Perimeter of the Base:
- Sides of the triangle: 10.2 m, 11.0 m, and 14.8 m
\[
\text{Perimeter of Base} = 10.2 + 11.0 + 14.8 = 36.0 \, \text{m}
\]

4. Calculate the Surface Area:
\[
\text{Surface Area} = (\text{Perimeter of Base} \times l) + (2 \times \text{Area of Base})
\]
\[
\text{Surface Area} = (36.0 \times 19.8) + (2 \times 56.1)
\]
\[
\text{Surface Area} = 712.8 + 112.2 = 825.0 \, \text{m}^2
\]

#### Problem 4:
- Dimensions:
- Base of triangle (\( b \)): 30.4 ft
- Height of triangle (\( h \)): 28.8 ft
- Prism Length (\( l \)): 20.6 ft

1. Calculate the Area of the Base:
\[
\text{Area of Base} = 0.5 \times b \times h = 0.5 \times 30.4 \times 28.8 = 0.5 \times 876.48 = 438.24 \, \text{ft}^2
\]

2. Calculate the Volume:
\[
V = \text{Area of Base} \times l = 438.24 \times 20.6 = 9030.624 \, \text{ft}^3
\]

3. Calculate the Perimeter of the Base:
- Sides of the triangle: 30.4 ft, 28.8 ft, and 32.6 ft
\[
\text{Perimeter of Base} = 30.4 + 28.8 + 32.6 = 91.8 \, \text{ft}
\]

4. Calculate the Surface Area:
\[
\text{Surface Area} = (\text{Perimeter of Base} \times l) + (2 \times \text{Area of Base})
\]
\[
\text{Surface Area} = (91.8 \times 20.6) + (2 \times 438.24)
\]
\[
\text{Surface Area} = 1891.08 + 876.48 = 2767.56 \, \text{ft}^2
\]

Final Answers:

1. Volume: \( 13347.636 \, \text{in}^3 \)
Surface Area: \( 4108.22 \, \text{in}^2 \)

2. Volume: \( 8120.752 \, \text{mm}^3 \)
Surface Area: \( 2731.28 \, \text{mm}^2 \)

3. Volume: \( 1110.18 \, \text{m}^3 \)
Surface Area: \( 825.0 \, \text{m}^2 \)

4. Volume: \( 9030.624 \, \text{ft}^3 \)
Surface Area: \( 2767.56 \, \text{ft}^2 \)

\boxed{
\begin{array}{ll}
\text{1. Volume: } 13347.636 \, \text{in}^3 & \text{Surface Area: } 4108.22 \, \text{in}^2 \\
\text{2. Volume: } 8120.752 \, \text{mm}^3 & \text{Surface Area: } 2731.28 \, \text{mm}^2 \\
\text{3. Volume: } 1110.18 \, \text{m}^3 & \text{Surface Area: } 825.0 \, \text{m}^2 \\
\text{4. Volume: } 9030.624 \, \text{ft}^3 & \text{Surface Area: } 2767.56 \, \text{ft}^2 \\
\end{array}
}