solid geometry

Question

A hut is in the shape of hemisphere surmounted on a cylinder the diameter of the base is 42m what is the area of the canvas required to cover the hut , if the maximum height of the hut is 24

 

(a)6545

(b)3698

(c)4000

(d)3168

Answer 1

If the diameter of the base is 42m, that would apply to both the cylinder and the hemisphere.  That means that the radius of of the hemisphere is 21m.

 

The "maximum height of the hut" must refer to the distance from the floor (bottom of the cylinder) to the top of the hemisphere.  It's made of of two distances -- the distance from the floor to the top of the cylinder (i.e., the height of the cylinder), and the distance from the top of the cylinder to the top of the hut.  The total height is 24, and the distance from the top of the cylinder to the top of the hut is the same as the radius of the hemisphere, 21.  So the height of the cylinder is 3.

 

Now we have everything we need.

 

The only part of the cylinder that needs to be covered with canvas is its side, since the bottom is next to the ground, and the top is covered by the hemisphere.  The area of that section is called the "lateral" surface area.  That's equal to 2πrh = 2 * π * 21 * 3 = 395.84.

 

The surface area of the hemisphere is 1/2 the surface area that the whole sphere would have, which is 4πr² -- so the surface area of the hemisphere is 2πr² = 2 * π * 21² = 2770.88.

 

So the total surface area that needs to be covered (i.e., the area of the canvas) is 395.84 + 2770.88 = 3166.72.  That's very close to answer (d) -- the difference must be due to differences in rounding.