Answer to Question #152774 in Geometry for Hardy

Question

The diagram shows a solid formed by joining a hemisphere, of radius r cm, to a cylinder, of a radius r cm and height h cm.

The total height of the solid is 18 cm and the surface area is 205π cm^2.

Find the value of r and the value of h.

Answer 1

Step 1: We have been given the total height of the solid = 18cm

The total surface area of the solid =205πcm² .... eq. (1)

The total surface area of the solid = Curved surface area of hemisphere + surface area of the cylinder

= 2πr² + πr² +2πrh ........ eq. (2)

On equating eq. 1 and eq. 2 we get

3πr² + 2πrh = 205π

3r² + 2rh = 205 ..........eq. (3)

Step 2: Since h + r = 18cm (the given total height of the solid)

h = 18 - r #

On putting the value of h = 18 - r in eq. 3, we get

3r² + 2r (18-r) = 205

3r² + 36r - 2r² = 205

r² + 36r - 205 = 0

Solving the quadratic equation

(r + 41) (r - 5)=0

Here we neglect r =-41 and we use r = 5

Step 3: Now substituting the value of r to eq. (3). hence:

h = 18 - 5 = 13 cm

Thus the radius and height of the soild are 5cm and 13 cm respectively