Answer:
Long solution
Explanation:
(a) Let the height of the cone be h cm. The volume of the hemisphere is given by (1/2)(4/3)πr³ = (2/3)πr³. The volume of the solid is the sum of the volumes of the hemisphere and the cone, which is (2/3)πr³ + (1/3)πr²h. Since the volume of the hemisphere is one third of the volume of the solid, we have:
(2/3)πr³ = (1/3)πr²h
Simplifying, we get:
2r = h
Therefore, h is expressed in terms of r as h = 2r.
(b) The slant height of the cone can be found using the Pythagorean theorem. Let l be the slant height, then we have:
l² = r² + h²
Substituting h = 2r, we get:
l² = r² + (2r)² = 5r²
Taking the square root of both sides, we get:
l = r√5
Since k is an integer, we can write:
l = Vk cm, where k is an integer
Comparing the two expressions, we get:
Vk = r√5
Therefore, the value of k is k = ⌊r√5⌋, where ⌊x⌋ denotes the largest integer less than or equal to x.
(c) The total surface area of the solid is the sum of the curved surface area of the cone, the curved surface area of the hemisphere, and the area of the circular base of the cone. We have:
Curved surface area of the cone = πr l = πr(r√5) = πr²√5
Curved surface area of the hemisphere = 2πr²
Area of the circular base of the cone = πr²
Therefore, the total surface area of the solid, in cm², is given by:
πr²√5 + 2πr² + πr² = (πr²)(√5 + 3)