Geometric Properties : Volume and Surface Area

(31) A cylindrical container has a diameter of 10 cm and a height of 18 cm. Calculate its volume.

• (a) 900π cm³
• (b) 450π cm³
• (c) 600π cm³
• (d) 750π cm³

Answer

Correct Answer: (b) 450π cm³
Explanation: The volume \( V \) of a cylindrical container with diameter \( d \) and height \( h \) is given by \( V = \frac{1}{4}\pi d^2 h \).
\[ V = \frac{1}{4}\pi \times 10^2 \times 18 = 450\pi \, cm^3 \]

(32) A cone has a base radius of 9 cm and a slant height of 15 cm. Calculate its volume.

• (a) 945π cm³
• (b) 675π cm³
• (c) 1350π cm³
• (d) 675π cm³

Answer

Correct Answer: (d) 675π 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.
\[ V = \frac{1}{3}\pi \times 9^2 \times \frac{15}{3} = 675\pi \, cm^3 \]

(33) A rectangular box has dimensions of 7 cm, 5 cm, and 10 cm. Calculate its total surface area.

• (a) 340 cm²
• (b) 400 cm²
• (c) 310 cm²
• (d) 450 cm²

Answer

Correct Answer: (c) 310 cm²
Explanation: The total surface area \( A \) of a rectangular box with dimensions \( l \), \( w \), and \( h \) is given by \( A = 2lw + 2wh + 2lh \).
\[ A = 2 \times 7 \times 5 + 2 \times 5 \times 10 + 2 \times 7 \times 10 = 310 \, cm^2 \]

(34) 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 \( V = \frac{4}{3}\pi r^3 \).
\[ 288\pi = \frac{4}{3}\pi r^3 \]
\[ r^3 = \frac{288\pi}{4\pi} \]
\[ r^3 = 72 \]
\[ r = \sqrt[3]{72} = 4 \, cm \]

(35) A triangular prism has a base with sides of length 6 cm each, a height of 8 cm, and a length of 10 cm. Calculate its total surface area.

• (a) 320 cm²
• (b) 360 cm²
• (c) 400 cm²
• (d) 440 cm²

Answer

Correct Answer: (c) 400 cm²
Explanation: The total surface area \( A \) of a triangular prism with base area \( B \) and height \( h \) is given by \( A = B + 2lw + 2lh + 2wh \).
\[ B = \frac{1}{2} \times 6 \times 8 = 24 \, cm^2 \]
\[ A = 24 + 2 \times 6 \times 10 + 2 \times 8 \times 10 + 2 \times 6 \times 8 = 400 \, cm^2 \]