Understanding how to calculate the surface areas and volumes of compound solids and parts of solids is crucial for solving real-world problems. This section will guide you through the process of determining these measurements for various compound solids, including scenarios such as half a sphere. We will focus on the application of formulas and solving example problems to equip you with the skills necessary to tackle exam questions effectively.

Introduction to Compound Solids

Compound solids are figures made up of two or more simple geometric shapes. The challenge in dealing with compound solids lies in breaking them down into their simpler components, for which surface area and volume formulas are readily available. This approach allows for the accurate calculation of the total surface area and volume of the compound solid.

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Example: Surface Area of a Hemisphere Attached to a Cylinder

Problem: A solid is composed of a cylinder with a radius of 5 cm and a height of 10 cm, with a hemisphere of the same radius attached to one of its circular faces. Calculate the total surface area of the solid.

Solution:

• Cylinder Surface Area (excluding the base attached to the hemisphere):

• Hemisphere Surface Area:

• Total Surface Area:

Volume of Compound Solids

The volume of a compound solid is found by calculating the volume of each basic shape and then summing these volumes.

Example: Volume of the Same Hemisphere and Cylinder Solid

Problem: Using the same solid from the previous example, calculate its total volume.

Solution:

• Cylinder Volume:

• Hemisphere Volume:

• Total Volume:

Practice Problems

Problem 1: Cone Topped with a Hemisphere

Question: A newly designed ice cream cone where the cone itself is topped with a perfectly spherical scoop of ice cream. The cone part of this ice cream cone has a height of 8 cm and a base with a radius of 3 cm. The scoop of ice cream, which fits exactly on top of the cone, forms a perfect hemisphere with a radius that matches the cone's base radius of 3 cm. Calculate the total surface area and volume of this delicious ice cream cone, taking into account both the cone and the hemisphere on top.

Solution:

• Cone Surface Area and Volume:

• Hemisphere Surface Area and Volume:

• Total Surface Area: $137.07 \, \text{cm}^2$

• Total Volume: $131.95 \, \text{cm}^3$

Problem 2: Cuboid with an Attached Cylinder

Question: A solid is formed by attaching a cylinder (radius = 3 cm, height = 7 cm) to one face of a cuboid (length = 10 cm, width = 6 cm, height = 7 cm). Calculate the total surface area and volume of this solid.

Solution:

• Cuboid Surface Area and Volume:
  • Surface Area: $2(lw + lh + wh)$ where $l = 10 \, \text{cm}$, $w = 6 \, \text{cm}$, and $h = 7 \, \text{cm}$
  • Volume: $l \times w \times h$

• Cylinder Surface Area and Volume (excluding the circular base attached to the cuboid):
  • Surface Area: $2\pi rh + \pi r^2$ (adding the area of the circular base that is visible).
  • Volume: $\pi r^2h$ where $r = 3 \, \text{cm}$ and $h = 7 \, \text{cm}$.

• Adjustments for Total Surface Area: Subtract the area of the cuboid face that the cylinder covers (one $lw$ face), then add the cylinder's surface area, including the area of its base that's visible.

• Total Volume Calculation: Simply add the volumes of the cuboid and the cylinder.

Given the values:

• $l = 10 \, \text{cm}, w = 6 \, \text{cm}, h_{\text{cuboid}} = 7 \, \text{cm}$

• $r = 3 \, \text{cm}, h_{\text{cylinder}} = 7 \, \text{cm}$