## Unit 11 Volume and Surface Area Homework 3 Answer Key

### Introduction

Unit 11 of your math course focuses on volume and surface area. In Homework 3, you have been given a set of problems to solve. This article serves as an answer key to help you check your work and understand the concepts better. Let's dive into the solutions!

### Problem 1: Finding the Volume of a Rectangular Prism

To find the volume of a rectangular prism, you need to multiply its length, width, and height. In this problem, the length is 5 cm, the width is 3 cm, and the height is 4 cm. Plugging these values into the volume formula, we get:

Volume = Length × Width × Height

Volume = 5 cm × 3 cm × 4 cm

Volume = 60 cm³

### Problem 2: Calculating the Surface Area of a Cylinder

The surface area of a cylinder can be found by adding the areas of its two circular bases and the lateral surface area. In this problem, the radius of the cylinder is 7 cm and its height is 10 cm. Let's calculate the surface area step by step:

Step 1: Calculate the area of each circular base:

Area of a circle = π × radius²

Area of each base = π × 7 cm²

Area of each base = 49π cm²

Step 2: Calculate the lateral surface area:

Lateral surface area = 2 × π × radius × height

Lateral surface area = 2 × π × 7 cm × 10 cm

Lateral surface area = 140π cm²

Step 3: Add the areas together:

Surface area = 2 × (Area of each base) + (Lateral surface area)

Surface area = 2 × (49π cm²) + (140π cm²)

Surface area = 238π cm²

### Problem 3: Finding the Volume of a Cone

The volume of a cone can be calculated by multiplying the area of its base (a circle) with its height and dividing the result by 3. In this problem, the radius of the cone's base is 6 cm and its height is 8 cm. Let's find the volume:

Step 1: Calculate the area of the base:

Area of a circle = π × radius²

Area of the base = π × 6 cm²

Area of the base = 36π cm²

Step 2: Calculate the volume:

Volume = (Area of the base × Height) ÷ 3

Volume = (36π cm² × 8 cm) ÷ 3

Volume = 96π cm³

### Problem 4: Calculating the Surface Area of a Sphere

The surface area of a sphere can be found by multiplying 4π with the square of its radius. In this problem, the radius of the sphere is 5 cm. Let's calculate the surface area:

Surface area = 4π × radius²

Surface area = 4π × 5 cm²

Surface area = 100π cm²

### Problem 5: Finding the Volume of a Triangular Prism

The volume of a triangular prism can be calculated by multiplying the area of its triangular base with its height. In this problem, the base of the prism is an equilateral triangle with side length 8 cm, and the height is 12 cm. Let's find the volume:

Step 1: Calculate the area of the base:

Area of an equilateral triangle = √3 × (side length)² ÷ 4

Area of the base = √3 × (8 cm)² ÷ 4

Area of the base = √3 × 64 cm² ÷ 4

Area of the base = 16√3 cm²

Step 2: Calculate the volume:

Volume = (Area of the base × Height)

Volume = (16√3 cm² × 12 cm)

Volume = 192√3 cm³

### Problem 6: Calculating the Surface Area of a Rectangular Prism

The surface area of a rectangular prism can be found by adding the areas of all its faces. In this problem, the length is 6 cm, the width is 4 cm, and the height is 5 cm. Let's calculate the surface area step by step:

Step 1: Calculate the area of each face:

Area of a rectangle = Length × Width

Area of the top and bottom faces = 6 cm × 4 cm

Area of the top and bottom faces = 24 cm²

Area of the side faces = 6 cm × 5 cm

Area of the side faces = 30 cm²

Step 2: Add the areas together:

Surface area = 2 × (Area of the top and bottom faces) + 2 × (Area of the side faces)

Surface area = 2 × (24 cm²) + 2 × (30 cm²)

Surface area = 108 cm²

### Problem 7: Finding the Volume of a Cylinder

The volume of a cylinder can be calculated by multiplying the area of its circular base with its height. In this problem, the radius of the cylinder is 3 cm and its height is 10 cm. Let's find the volume:

Step 1: Calculate the area of the base:

Area of a circle = π × radius²

Area of the base = π × 3 cm²

Area of the base = 9π cm²

Step 2: Calculate the volume:

Volume = (Area of the base × Height)

Volume = (9π cm² × 10 cm)

Volume = 90π cm³

### Problem 8: Calculating the Surface Area of a Cone

The surface area of a cone can be found by adding the area of its base (a circle) with the area of its lateral surface. In this problem, the radius of the cone's base is 4 cm and its slant height is 6 cm. Let's calculate the surface area step by step:

Step 1: Calculate the area of the base:

Area of a circle = π × radius²

Area of the base = π × 4 cm²

Area of the base = 16π cm²

Step 2: Calculate the lateral surface area:

Lateral surface area = π × radius × slant height

Lateral surface area = π × 4 cm × 6 cm

Lateral surface area = 24π cm²

Step 3: Add the areas together:

Surface area = (Area of the base) + (Lateral surface area)

Surface area = (16π cm²) + (24π cm²)

Surface area = 40π cm²

### Problem 9: Finding the Volume of a Sphere

The volume of a sphere can be calculated by multiplying 4/3π with the cube of its radius. In this problem, the radius of the sphere is 7 cm. Let's calculate the volume:

Volume = (4/3π) × radius³

Volume = (4/3π) × (7 cm)³

Volume = (4/3π) × 343 cm³

Volume = 1436.76 cm³

### Problem 10: Calculating the Surface Area of a Triangular Prism

The surface area of a triangular prism can be found by adding the areas of its two triangular bases and the areas of its three rectangular faces. In this problem, the base of the prism is an equilateral triangle with side length 5 cm, and the height is 8 cm. Let's calculate the surface area step by step:

Step 1: Calculate the area of each triangular base:

Area of an equilateral triangle = √3 × (side length)² ÷ 4

Area of each base = √3 × (5 cm)² ÷ 4

Area of each base = √3 × 25 cm² ÷ 4