Final answer:
To find the slant height and C.S.A of the cone, we need to use the formula for the volume of a cone. Given the volume and the diameter of the base, we can calculate the radius and height of the cone. Then, using the Pythagorean theorem, we can find the slant height. Finally, the C.S.A can be found using the formula C.S.A = π * r * l, where r is the radius and l is the slant height. Hence, C.S.A ≈ 1451.37 cm².
Step-by-step explanation:
To find the slant height and C.S.A of the cone, we need to use the formula for the volume of a cone, which is:
V = 1/3 * π * r^2 * h, where V is the volume, π is a mathematical constant approximated to 3.142, r is the radius of the base, and h is the height of the cone.
In this case, we are given the diameter of the base, so we need to divide it by 2 to get the radius. Given that the diameter is 42 cm, the radius will be 21 cm.
Now, let's solve for h. We have the volume, which is 12936 cm^3.
Plugging in the values, we get: 12936 = 1/3 * 3.142 * (21)^2 * h.
Simplifying, we get: 12936 = 3.142 * 441 * h. Dividing both sides by (3.142 * 441), we find that h ≈ 9.
Finally, we can find the slant height of the cone using the Pythagorean theorem.
The slant height (l) can be found using the formula l = √(r^2 + h^2).
Plugging in the values, we get: l = √(21^2 + 9^2) ≈ √{441 + 81} ≈ √522 ≈ 22.87 cm.
So, the slant height of the cone is approximately 22.87 cm and the C.S.A (Curved Surface Area) can be found using the formula C.S.A = π * r * l, where r is the radius and l is the slant height.
Plugging in the values, we get: C.S.A = 3.142 * 21 * 22.87 ≈ 1451.37 cm².