Understanding Volume and TSA of 3D Shapes

Question 1: A cube has a side length of 4 cm. What is its volume and total surface area?

Answer:
To calculate the volume (V) of a cube, we use the formula:
V = side^3
For a cube with a side length of 4 cm:
V = 4^3 = 64 cubic centimeters.

To calculate the total surface area (TSA) of a cube, we use the formula:
TSA = 6 × side^2
TSA = 6 × 4^2 = 6 × 16 = 96 square centimeters.

So, the volume is 64 cm³ and the total surface area is 96 cm².

Question 2: A cylindrical water tank has a radius of 3 m and a height of 5 m. What is the volume of the tank and its total surface area?

Answer:
To find the volume (V) of a cylinder, we use the formula:
V = π × radius^2 × height
Using π as approximately 3.14:
V = 3.14 × 3^2 × 5 = 3.14 × 9 × 5 = 141.3 cubic meters.

To calculate the total surface area (TSA) of the cylinder, we use the formula:
TSA = 2 × π × radius × (radius + height)
TSA = 2 × 3.14 × 3 × (3 + 5) = 2 × 3.14 × 3 × 8 = 150.72 square meters.

Thus, the volume is approximately 141.3 m³ and the total surface area is approximately 150.72 m².

Question 3: A rectangular box (cuboid) measures 10 cm in length, 6 cm in width, and 4 cm in height. Calculate its volume and total surface area.

Answer:
To find the volume (V) of a cuboid, we use the formula:
V = length × width × height
V = 10 × 6 × 4 = 240 cubic centimeters.

To calculate the total surface area (TSA) of the cuboid, we use the formula:
TSA = 2 × (length × width + length × height + width × height)
TSA = 2 × (10 × 6 + 10 × 4 + 6 × 4) = 2 × (60 + 40 + 24) = 2 × 124 = 248 square centimeters.

Therefore, the volume is 240 cm³ and the total surface area is 248 cm².

Question 4: Imagine you are designing a gift box in the shape of a cube, and you want to wrap it in decorative paper. If the side length of the box is 5 cm, how much wrapping paper will you need?

Answer:
The amount of wrapping paper needed is equal to the total surface area of the cube. Using the formula:
TSA = 6 × side^2
TSA = 6 × 5^2 = 6 × 25 = 150 square centimeters.

You will need 150 cm² of wrapping paper to cover the gift box.

Question 5: A cylindrical storage tank has a height of 10 m and a radius of 2 m. If you want to paint the outside of the tank, how much paint will you need if 1 liter of paint covers 10 m²?

Answer:
First, calculate the total surface area (TSA) for the cylinder:
TSA = 2 × π × radius × (radius + height)
TSA = 2 × 3.14 × 2 × (2 + 10) = 2 × 3.14 × 2 × 12 = 150.72 square meters.

To find out how much paint is needed:
150.72 m² ÷ 10 m²/liter = 15.072 liters.

Therefore, you will need approximately 15.1 liters of paint to cover the outside of the tank.

These questions incorporate real-life scenarios and progressively increase in complexity. They encourage problem-solving and application of formulas while providing a solid understanding of TSA and Volume calculations for cubes, cylinders, and cuboids.