Geometric Properties : Volume and Surface Area

(46) A cylindrical container has a diameter of 8 cm and a height of 14 cm. Calculate its total surface area.

• (a) 512π cm²
• (b) 376π cm²
• (c) 448π cm²
• (d) 420π cm²

Answer

Correct Answer: (b) 376π cm²
Explanation: The total surface area \( A \) of a cylindrical container with diameter \( d \) and height \( h \) is given by \( A = 2\pi r^2 + 2\pi rh \), where \( r \) is the radius.
\[ r = \frac{d}{2} = \frac{8}{2} = 4 \, cm \]
\[ A = 2\pi \times 4^2 + 2\pi \times 4 \times 14 = 376\pi \, cm^2 \]

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

• (a) 1190π cm³
• (b) 1870π cm³
• (c) 850π cm³
• (d) 2210π cm³

Answer

Correct Answer: (c) 850π 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.
\[ V = \frac{1}{3}\pi \times 10^2 \times \frac{17}{3} = 850\pi \, cm^3 \]

(48) A rectangular box has dimensions of 8 cm, 5 cm, and 6 cm. Calculate its total surface area.

• (a) 260 cm²
• (b) 244 cm²
• (c) 280 cm²
• (d) 312 cm²

Answer

Correct Answer: (b) 244 cm²
Explanation: The total surface area \( A \) of a rectangular box with dimensions \( l \), \( w \), and \( h \) is given by \( A = 2lw + 2wh + 2lh \).
\[ A = 2 \times 8 \times 5 + 2 \times 5 \times 6 + 2 \times 8 \times 6 = 244 \, cm^2 \]

(49) A sphere has a volume of 1256π cm³. Calculate its radius.

• (a) 7 cm
• (b) 8 cm
• (c) 9 cm
• (d) 10 cm

Answer

Correct Answer: (b) 8 cm
Explanation: The volume \( V \) of a sphere with radius \( r \) is given by \( V = \frac{4}{3}\pi r^3 \).
\[ 1256\pi = \frac{4}{3}\pi r^3 \]
\[ r^3 = \frac{1256\pi}{4\pi} \]
\[ r^3 = 314 \]
\[ r = \sqrt[3]{314} \approx 8 \, cm \]

(50) A triangular pyramid has a base with sides of length 12 cm each and a height of 15 cm. Calculate its total surface area.

• (a) 540 cm²
• (b) 576 cm²
• (c) 720 cm²
• (d) 648 cm²

Answer

Correct Answer: (c) 720 cm²
Explanation: The total surface area \( A \) of a triangular pyramid with base area \( B \) and height \( h \) is given by \( A = B + 2lw + 2lh + 2wh \).
\[ B = \frac{1}{2} \times 12 \times 15 = 90 \, cm^2 \]
\[ A = 90 + 2 \times 12 \times 15 + 2 \times 15 \times 15 + 2 \times 12 \times 15 = 720 \, cm^2 \]