Geometric Properties : Volume and Surface Area

(41) A cylindrical tank has a radius of 5 meters and a height of 16 meters. Calculate its volume.

• (a) 400π m³
• (b) 400π m³
• (c) 800π m³
• (d) 320π m³

Answer

Correct Answer: (b) 400π m³
Explanation: The volume \( V \) of a cylindrical tank with radius \( r \) and height \( h \) is given by \( V = \pi r^2 h \).
\[ V = \pi \times 5^2 \times 16 = 400\pi \, m^3 \]

(42) A cone has a base radius of 8 cm and a slant height of 15 cm. Calculate its volume.

• (a) 360π cm³
• (b) 480π cm³
• (c) 720π cm³
• (d) 480π cm³

Answer

Correct Answer: (d) 480π cm³
Explanation: The volume \( V \) of a cone with base radius \( r \) and slant height \( l \) is given by \( V = \frac{1}{3}\pi r^2 h \), where \( h \) is the height of the cone.
\[ V = \frac{1}{3}\pi \times 8^2 \times \frac{15}{3} = 480\pi \, cm^3 \]

(43) A rectangular box has dimensions of 6 cm, 7 cm, and 9 cm. Calculate its total surface area.

• (a) 324 cm²
• (b) 330 cm²
• (c) 360 cm²
• (d) 378 cm²

Answer

Correct Answer: (b) 330 cm²
Explanation: The total surface area \( A \) of a rectangular box with dimensions \( l \), \( w \), and \( h \) is given by \( A = 2lw + 2wh + 2lh \).
\[ A = 2 \times 6 \times 7 + 2 \times 7 \times 9 + 2 \times 6 \times 9 = 330 \, cm^2 \]

(44) A sphere has a volume of 288π cm³. Calculate its radius.

• (a) 6 cm
• (b) 4 cm
• (c) 8 cm
• (d) 5 cm

Answer

Correct Answer: (b) 4 cm
Explanation: The volume \( V \) of a sphere with radius \( r \) is given by \( V = \frac{4}{3}\pi r^3 \).
\[ 288\pi = \frac{4}{3}\pi r^3 \]
\[ r^3 = \frac{288\pi}{4\pi} \]
\[ r^3 = 72 \]
\[ r = \sqrt[3]{72} = 4 \, cm \]

(45) A triangular prism has a base with sides of length 5 cm each, a height of 9 cm, and a length of 12 cm. Calculate its total surface area.

• (a) 450 cm²
• (b) 480 cm²
• (c) 540 cm²
• (d) 600 cm²

Answer

Correct Answer: (c) 540 cm²
Explanation: The total surface area \( A \) of a triangular prism with base area \( B \) and height \( h \) is given by \( A = B + 2lw + 2lh + 2wh \).
\[ B = \frac{1}{2} \times 5 \times 9 = 22.5 \, cm^2 \]
\[ A = 22.5 + 2 \times 5 \times 12 + 2 \times 9 \times 12 + 2 \times 5 \times 9 = 540 \, cm^2 \]