Final answer:
To find the height and slant height of the cone, the volume of the original hollow sphere was equated to the volume of the cone. After calculations, the height of the cone is 14 cm and the slant height is approximately 14.56 cm.
Step-by-step explanation:
The question requires us to find the height and slant height of a cone formed from a melted hollow sphere. To solve this, we need to equate the volume of the original hollow sphere to the volume of the cone, since melting does not change the volume.
Step-by-step Solution:
The formula for the volume of a hollow sphere is V = ⅔/3π(R^3 - r^3), where R is the external radius and r is the internal radius.
In this case, R = 4 cm and r = 2 cm.
Plugging the values into the formula we get: V = ⅔/3π(4^3 - 2^3) = ⅔/3π(64 - 8) = ⅔/3π(56).
Now, the volume of a cone is given by V = 1/3πh(r^2), where h is the height and r is the base radius of the cone.
Since the base radius of the cone is 4 cm, we have: V = 1/3πh(4^2).
To find the height of the cone, we set the volumes equal to each other: ⅔/3π(56) = 1/3πh(16).
After simplifying, h = 14 cm. Thus, the height of the cone is 14 cm.
To find the slant height (l), we use the Pythagorean theorem: l^2 = r^2 + h^2. Here, r = 4 cm and h = 14 cm.
So, l^2 = 4^2 + 14^2 => l^2 = 16 + 196 => l^2 = 212. Taking the square root, we get l ≈ 14.56 cm.
Therefore, the answer is: Height = 14 cm and Slant height ≈ 14.56 cm.