Chapter 13 Surface Areas and Volumes

Case Study 1: Volume of a Cylinder

Case Description:
A water tank in the shape of a cylinder has a height of 5 meters and a radius of 2 meters. The tank is used to store water for the community. To determine how much water the tank can hold, we need to calculate the volume of the cylinder using the formula $V = \pi r^2 h$, where $r$ is the radius and $h$ is the height. The community is planning to fill the tank with water and wants to know the total capacity in cubic meters.

MCQs:

1. What is the radius of the cylindrical water tank?

   • A) 1 m
   • B) 2 m
   • C) 3 m
   • D) 5 m

2. What is the height of the water tank in meters?

   • A) 4 m
   • B) 5 m
   • C) 6 m
   • D) 7 m

3. Using $\pi \approx 3.14$, what is the volume of the water tank?

   • A) 31.4 m³
   • B) 62.8 m³
   • C) 25.12 m³
   • D) 40.84 m³

4. If the tank is filled to capacity, how many liters of water does it hold? (1 m³ = 1000 liters)

   • A) 31400 liters
   • B) 62800 liters
   • C) 25120 liters
   • D) 40840 liters

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Case Study 2: Surface Area of a Cone

Case Description:
A party hat is in the shape of a cone with a radius of 3 cm and a height of 12 cm. The party planner wants to decorate the hat and needs to know the surface area to purchase enough material. The surface area of a cone is calculated using the formula $SA = \pi r (r + l)$, where $l$ is the slant height of the cone, calculated as $l = \sqrt{r^2 + h^2}$.

MCQs:

1. What is the radius of the party hat in centimeters?

   • A) 2 cm
   • B) 3 cm
   • C) 4 cm
   • D) 5 cm

2. What is the height of the cone?

   • A) 10 cm
   • B) 11 cm
   • C) 12 cm
   • D) 13 cm

3. What is the slant height $l$ of the cone using the formula $l = \sqrt{r^2 + h^2}$?

   • A) 10 cm
   • B) 11 cm
   • C) 12 cm
   • D) 15 cm

4. Using $\pi \approx 3.14$, what is the total surface area of the cone?

   • A) 73.44 cm²
   • B) 94.25 cm²
   • C) 112.56 cm²
   • D) 150.76 cm²

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Case Study 3: Volume of a Sphere

Case Description:
A basketball is in the shape of a sphere with a radius of 12 cm. To understand how much air is required to inflate the basketball, the coach needs to calculate the volume using the formula $V = \frac{4}{3} \pi r^3$. This volume will help in determining the right amount of air for optimal performance.

MCQs:

1. What is the radius of the basketball in centimeters?

   • A) 10 cm
   • B) 11 cm
   • C) 12 cm
   • D) 13 cm

2. Using $\pi \approx 3.14$, what is the volume of the basketball?

   • A) 576 cm³
   • B) 904.32 cm³
   • C) 288 cm³
   • D) 1296 cm³

3. If the basketball is inflated to capacity, how much space does it occupy?

   • A) 900 cm³
   • B) 500 cm³
   • C) 1000 cm³
   • D) 800 cm³

4. How many basketballs of this size can fit into a container with a volume of 10000 cm³?

   • A) 10
   • B) 12
   • C) 11
   • D) 9

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Case Study 4: Surface Area of a Cylinder

Case Description:
A cylindrical gift box has a radius of 4 cm and a height of 10 cm. The gift box needs to be wrapped, and the wrapper's total surface area must be calculated. The surface area of a cylinder can be found using the formula $SA = 2\pi r (r + h)$, which accounts for both the curved surface area and the areas of the circular bases.

MCQs:

1. What is the radius of the cylindrical gift box in centimeters?

   • A) 2 cm
   • B) 3 cm
   • C) 4 cm
   • D) 5 cm

2. What is the height of the gift box?

   • A) 8 cm
   • B) 10 cm
   • C) 12 cm
   • D) 15 cm

3. Using $\pi \approx 3.14$, what is the total surface area of the gift box?

   • A) 150.8 cm²
   • B) 251.2 cm²
   • C) 301.6 cm²
   • D) 200 cm²

4. If the wrapper costs ₹2 per square centimeter, how much will it cost to wrap the gift box?

   • A) ₹350
   • B) ₹450
   • C) ₹500
   • D) ₹600

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Case Study 5: Volume of a Hemisphere

Case Description:
An aquarium in the shape of a hemisphere has a radius of 6 meters. The aquarium is designed for aquatic plants and fish, and to understand its capacity, the owner needs to calculate the volume of the hemisphere using the formula $V = \frac{2}{3} \pi r^3$. This will help in determining how much water is needed to fill the aquarium.

MCQs:

1. What is the radius of the hemisphere in meters?

   • A) 4 m
   • B) 5 m
   • C) 6 m
   • D) 7 m

2. Using $\pi \approx 3.14$, what is the volume of the hemisphere?

   • A) 50.24 m³
   • B) 113.04 m³
   • C) 78.24 m³
   • D) 226.08 m³

3. If the aquarium is filled to capacity, how many liters of water does it hold? (1 m³ = 1000 liters)

   • A) 50000 liters
   • B) 78000 liters
   • C) 113040 liters
   • D) 226080 liters

4. What would happen to the volume if the radius is increased to 8 meters?

   • A) The volume will decrease.
   • B) The volume will remain the same.
   • C) The volume will increase.
   • D) The volume will become negative.