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5. Using the following 3D shape, do each of the following: • Identify the 3D solid. Be as specific as possible. • Verify Euler's formula holds for the shape (show F + V = E + 2) • Calculate the surface area and show all work. • Calculate the volume and show all work. 8 cm 6 cm 12 cm

5. Using the following 3D shape, do each of the following: • Identify the 3D solid. Be as specific as possible. • Verify Euler's formula holds for the shape (show F + V = E + 2) • Calculate the surface area and show all work. • Calculate the volume and show all work. 8 cm 6 cm 12 cm

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.4: Plane Curves And Parametric Equations
Problem 39E
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### Mathematics Lesson: Working with 3D Shapes and Ratios #### Problem 5: Using the following 3D shape, do each of the following: - **Identify the 3D solid. Be as specific as possible.** - **Verify Euler's formula holds for the shape (show F + V = E + 2)** - **Calculate the surface area and show all work.** - **Calculate the volume and show all work.** ##### Diagram Description: - A 3D shape resembling a triangular prism is depicted. - The base of the triangle is 8 cm. - The height of the triangular face is 6 cm. - The length of the prism (distance between the triangular faces) is 12 cm. #### Solution Steps: 1. **Identify the 3D Solid:** - The shape is a triangular prism. 2. **Verify Euler's Formula:** - For a prism, Euler's formula (F + V = E + 2) should hold. - F (Faces) = 5 (3 rectangular faces and 2 triangular faces) - V (Vertices) = 6 (3 vertices per triangular face) - E (Edges) = 9 (3 per triangular face and 3 connecting the corresponding vertices of the triangular faces) Euler’s formula: 5 (F) + 6 (V) = 9 (E) + 2 --> 11 = 11 3. **Calculate the Surface Area:** - Surface Area = (Lateral Area) + (Area of 2 Bases) - Area of 2 triangular bases = 2 * (1/2 * base * height) = 2 * (1/2 * 8 cm * 6 cm) = 2 * 24 cm² = 48 cm² - Lateral Area = (Perimeter of the triangle * length of the prism) - For the triangular face: Perimeter = 8 cm + 6 cm + hypotenuse (10 cm due to Pythagorean theorem) = 24 cm - Lateral Area = 24 cm * 12 cm = 288 cm² - Total Surface Area = 48 cm² + 288 cm² = 336 cm² 4. **Calculate the Volume:** - Volume = Base Area * length = (1/2 * base * height
Transcribed Image Text:### Mathematics Lesson: Working with 3D Shapes and Ratios #### Problem 5: Using the following 3D shape, do each of the following: - **Identify the 3D solid. Be as specific as possible.** - **Verify Euler's formula holds for the shape (show F + V = E + 2)** - **Calculate the surface area and show all work.** - **Calculate the volume and show all work.** ##### Diagram Description: - A 3D shape resembling a triangular prism is depicted. - The base of the triangle is 8 cm. - The height of the triangular face is 6 cm. - The length of the prism (distance between the triangular faces) is 12 cm. #### Solution Steps: 1. **Identify the 3D Solid:** - The shape is a triangular prism. 2. **Verify Euler's Formula:** - For a prism, Euler's formula (F + V = E + 2) should hold. - F (Faces) = 5 (3 rectangular faces and 2 triangular faces) - V (Vertices) = 6 (3 vertices per triangular face) - E (Edges) = 9 (3 per triangular face and 3 connecting the corresponding vertices of the triangular faces) Euler’s formula: 5 (F) + 6 (V) = 9 (E) + 2 --> 11 = 11 3. **Calculate the Surface Area:** - Surface Area = (Lateral Area) + (Area of 2 Bases) - Area of 2 triangular bases = 2 * (1/2 * base * height) = 2 * (1/2 * 8 cm * 6 cm) = 2 * 24 cm² = 48 cm² - Lateral Area = (Perimeter of the triangle * length of the prism) - For the triangular face: Perimeter = 8 cm + 6 cm + hypotenuse (10 cm due to Pythagorean theorem) = 24 cm - Lateral Area = 24 cm * 12 cm = 288 cm² - Total Surface Area = 48 cm² + 288 cm² = 336 cm² 4. **Calculate the Volume:** - Volume = Base Area * length = (1/2 * base * height
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