Question

Volume of a cone rh, curved surface area of a cone = xr!] [Volume of a spheresurface area of a sphere 4ar']

The solid is formed from a hemisphere of radius rcm fixed to a cone of radius rcm and height hem. The volume of the hemisphere is one third of the volume of the solid.

(a) Find h in terms of r

(b) The slant height of the cone can be written as Vk cm, where k is an integer.

Find the value of k

(c) Find an expressionin terms of r and x, for the total surface area, in cm², of the solid

Answer 1

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)