(a) To find the slant height of the cone, we can use the Pythagorean theorem. The slant height (l) of a cone is the hypotenuse of a right triangle formed by the height (h) and the radius (r). In this case, the height (h) of the cone is given as 10 cm and the radius (r) is given as 5 cm.
Using the Pythagorean theorem:
l² = r² + h²
l² = 5² + 10²
l² = 25 + 100
l² = 125
Taking the square root of both sides:
l = √125
l ≈ 11.18 cm
Therefore, the slant height of the cone is approximately 11.18 cm.
(b) The curved surface area (CSA) of a cylinder is given by the formula:
CSA of cylinder = 2πrh
Where r is the radius of the cylinder's base and h is the height of the cylinder.
The curved surface area (CSA) of a cone is given by the formula:
CSA of cone = πrl
Where r is the radius of the cone's base and l is the slant height of the cone.
In this case, the radius (r) for both the cylinder and the cone is 5 cm, and the height (h) for the cylinder is 10 cm.
CSA of cylinder = 2π(5)(10) = 100π cm²
CSA of cone = π(5)(11.18) = 175.93π cm²
To find the ratio of the curved surface area of the cylinder to that of the cone, we divide the CSA of the cylinder by the CSA of the cone:
Ratio = (CSA of cylinder) / (CSA of cone)
Ratio = (100π) / (175.93π)
Ratio ≈ 0.569
Therefore, the ratio of the curved surface area of the cylinder to that of the cone is approximately 0.569.