Directions:

From a solid cylinder of height 30 cm and radius 7 cm, a conical cavity of height 24 cm and of base radius 7 cm is drilled out.

Based on this information, answer the following questions:

65. The volume of the remaining solid is :

(a) 2856 cm³
(b) 3388 cm³
(c) 3672 cm³
(d) 4620 cm³

66. The total surface area of the remaining solid is :

(a) 1870 cm²
(b) 2024 cm²
(c) 2178 cm²
(d) 2332 cm²

67. The slant height of the cut out cone is :

(a) 18 cm
(b) 25 cm
(c) 26 cm
(d) 32 cm

68. The total surface area of the cut out cone is :

(a) 550 cm²
(b) 704 cm²
(c) 858 cm²
(d) 616 cm²

65. Radius of solid cylinder = Radius of cone = r = 7 cm

Height of the cylinder, H = 30 cm

Height of cone, h = 24 cm

Volume of remaining solid = Volume of cylinder - Volume of cone

$= π\text{r}^2\text{H} - \dfrac{1}{3}π\text{r}^2\text{h} \\[1em] = π\text{r}^2(\text{H} - \dfrac{\text{h}}{3}) \\[1em] = \dfrac{22}{7} \times 7^2 \times (30 - \dfrac{24}{3}) \\[1em] = \dfrac{22}{7} \times 49 \times (30 - 8) \\[1em] = 22 \times 7 \times 22 \\[1em] = 3388 \text{ cm}^3.$

Hence, Option (b) is the correct option.

66. Slant height of cone, l = $\sqrt{\text{h}^2 + \text{r}^2} = \sqrt{24^2 + 7^2} = \sqrt{576 + 49} = \sqrt{625}$ = 25 cm

Total surface area of reamining solid = Curved surface area of cylinder + Area of base of cylinder + Curved surface area of cone

= 2πrH + πr² + πrl

= πr(2H + r + l)

= $\dfrac{22}{7}$ × 7 (2 × 30 + 7 + 25)

= 22 × (60 + 32)

= 22 × 92

= 2024 cm².

Hence, Option (b) is the correct option.

67. Slant height of cone, l = $\sqrt{\text{h}^2 + \text{r}^2} = \sqrt{24^2 + 7^2} = \sqrt{576 + 49} = \sqrt{625}$ = 25 cm

Hence, Option (b) is the correct option.

68. Total surface area of cut out cone = πr² + πrl

= πr(r + l)

= $\dfrac{22}{7}$ × 7 × (7 + 25)

= 22 × 32

= 704 cm²

Hence, Option (b) is the correct option.