Geometric Properties : Volume and Surface Area

(16) A rectangular prism has dimensions of 4 cm, 7 cm, and 9 cm. Calculate its total surface area.

• (a) 284 cm²
• (b) 316 cm²
• (c) 266 cm²
• (d) 298 cm²

Answer

Correct Answer: (c) 266 cm²
Explanation: The total surface area \( A \) of a rectangular prism with dimensions \( l \), \( w \), and \( h \) is given by \( A = 2lw + 2wh + 2lh \).
\[ A = 2 \times 4 \times 7 + 2 \times 7 \times 9 + 2 \times 4 \times 9 = 266 \, cm^2 \]

(17) A cone has a base radius of 8 cm and a slant height of 10 cm. Calculate its volume.

• (a) 160π cm³
• (b) 213.33π cm³
• (c) 240π cm³
• (d) 170π cm³

Answer

Correct Answer: (b) 213.33π cm³
Explanation: The volume \( V \) of a cone with 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.
\[ V = \frac{1}{3}\pi \times 8^2 \times \frac{10}{3} = 213.33\pi \, cm^3 \]

(18) A sphere has a volume of 36π cm³. Calculate its radius.

• (a) 3 cm
• (b) 4 cm
• (c) 2 cm
• (d) 5 cm

Answer

Correct Answer: (c) 2 cm
Explanation: The volume \( V \) of a sphere with radius \( r \) is given by \( V = \frac{4}{3}\pi r^3 \).
\[ \frac{4}{3}\pi r^3 = 36\pi \]
\[ r^3 = \frac{36\pi \times 3}{4\pi} \]
\[ r^3 = 27 \]
\[ r = \sqrt[3]{27} = 3 \, cm \]

(19) A cylindrical tank has a diameter of 6 meters and a height of 15 meters. Calculate the total surface area of the tank.

• (a) 340π m²
• (b) 270π m²
• (c) 300π m²
• (d) 320π m²

Answer

Correct Answer: (c) 300π m²
Explanation: The total surface area \( A \) of a cylinder with diameter \( d \) and height \( h \) is given by \( A = 2\pi r^2 + 2\pi rh \), where \( r \) is the radius of the cylinder.
\[ A = 2\pi \times \left(\frac{6}{2}\right)^2 + 2\pi \times \frac{6}{2} \times 15 = 300\pi \, m^2 \]

(20) A pyramid has a base with sides of length 12 cm each and a height of 8 cm. Calculate the volume of the pyramid.

• (a) 256 cm³
• (b) 256 cm³
• (c) 192 cm³
• (d) 224 cm³

Answer

Correct Answer: (b) 256 cm³
Explanation: The volume \( V \) of a pyramid with base area \( B \) and height \( h \) is given by \( V = \frac{1}{3}Bh \).
\[ B = \frac{1}{2} \times 12 \times 8 = 48 \, cm^2 \]
\[ V = \frac{1}{3} \times 48 \times 8 = 256 \, cm^3 \]