First, we need to find the slant heights of both the original cone and the conical cavity. The slant height (l) can be calculated using the Pythagorean theorem: l = √(r² + h²), where r is the radius and h is the height. For the larger cone, r_1 = 12 cm and h_1 = 16 cm. For the smaller conical cavity, r_2 = 4 cm and h_2 = 3 cm.

First, we need to find the slant heights of both the original cone and the conical cavity. The slant height (l) can be calculated using the Pythagorean theorem: l = √(r² + h²), where r is the radius and h is the height. For the larger cone, r_1 = 12 cm and h_1 = 16 cm. For the smaller conical cavity, r_2 = 4 cm and h_2 = 3 cm.

Substitute the given values into the slant height formula. For the larger cone, l_1 = √(12² + 16²) = √(144 + 256) = √400 = 20 cm. For the smaller cone, l_2 = √(4² + 3²) = √(16 + 9) = √25 = 5 cm.

Substitute the given values into the slant height formula. For the larger cone, l_1 = √(12² + 16²) = √(144 + 256) = √400 = 20 cm. For the smaller cone, l_2 = √(4² + 3²) = √(16 + 9) = √25 = 5 cm.

Now, substitute the calculated radii and slant heights into the area formulas. The area of the base ring is π (12² - 4²) = π (144 - 16) = 128π cm². The curved surface area of the large cone is π (12)(20) = 240π cm². The curved surface area of the small cone is π (4)(5) = 20π cm².

Now, substitute the calculated radii and slant heights into the area formulas. The area of the base ring is π (12² - 4²) = π (144 - 16) = 128π cm². The curved surface area of the large cone is π (12)(20) = 240π cm². The curved surface area of the small cone is π (4)(5) = 20π cm².

Adding the three areas together: 128π + 240π + 20π = 388π cm². If we approximate π ≈ 3.14, the total surface area is approximately 388 × 3.14 = 1218.32 cm².

From a solid cone, whose height is 16 cm and radius 12 cm, and conical cavity of height 3 cm and base radius 4 cm is hollowed out such that the bases of the cones form concentric circles. Find the total surface area of the remaining solid.

Answer: $388\pi \text{ cm}^2$

Step-by-step solution

Step 1: Calculate slant heights

First, we need to find the slant heights of both the original cone and the conical cavity. The slant height ( $l$ ) can be calculated using the Pythagorean theorem: $l = \sqrt{r^2 + h^2}$ , where $r$ is the radius and $h$ is the height. For the larger cone, $r_1 = 12 \text{ cm}$ and $h_1 = 16 \text{ cm}$ . For the smaller conical cavity, $r_2 = 4 \text{ cm}$ and $h_2 = 3 \text{ cm}$ .

Step 2: Substitute values and calculate slant heights

Substitute the given values into the slant height formula. For the larger cone, $l_1 = \sqrt{12^2 + 16^2} = \sqrt{144 + 256} = \sqrt{400} = 20 \text{ cm}$ . For the smaller cone, $l_2 = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \text{ cm}$ .

Step 3: Calculate areas

The total surface area of the remaining solid consists of three parts: the area of the circular base ring (the base of the large cone minus the base of the small cone), the curved surface area of the large cone, and the curved surface area of the hollowed-out small cone. The formula for the area of a circle is $\pi r^2$ and for the curved surface area of a cone is $\pi r l$ .

Step 4: Substitute values and calculate areas

Now, substitute the calculated radii and slant heights into the area formulas. The area of the base ring is $\pi (12^2 - 4^2) = \pi (144 - 16) = 128\pi \text{ cm}^2$ . The curved surface area of the large cone is $\pi (12)(20) = 240\pi \text{ cm}^2$ . The curved surface area of the small cone is $\pi (4)(5) = 20\pi \text{ cm}^2$ .

Step 5: Calculate total surface area

The total surface area of the remaining solid is the sum of these three calculated areas.

Step 6: Sum the areas

Adding the three areas together: $128\pi + 240\pi + 20\pi = 388\pi \text{ cm}^2$ . If we approximate $\pi \approx 3.14$ , the total surface area is approximately $388 \times 3.14 = 1218.32 \text{ cm}^2$ .

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