Problem Solving

A right circular cone is inscribed in a hemisphere so that the base of the cone coincides with the base of the hemisphere. What is the ratio of the height of the cone to the radius of the hemisphere?
A. 2: 1
B. (1/2): 1
C. 1: 1
D. √2: 1
E. √3: 1

Consider a hemisphere which is half of a ball.

If the base of the right circular coincides with the base of the hemisphere

then the height of the cone will be from the center of the hemisphere to the point where a straight line (at 90 degrees to the base) cuts the hemisphere

which is the radius of the hemisphere.

Answer should be V3 : 1 (E)

Please find my explanation in a diagramatic representation.

Vemuri wrote:Answer should be V3 : 1 (E)

Please find my explanation in a diagramatic representation.

You may please have one more chance to refine your answer, if the two bases coincide, as the case is here, base radius of cone is same as the radius of hemisphere; and height of such cone is same as the radius of the hemisphere.

What is your answer now?

I understand your question. But, how are you saying that "the two bases coincide, as the case is here, base radius of cone is same as the radius of hemisphere; and height of such cone is same as the radius of the hemisphere".

Looks like I am missing a concept here. Can you please explain?

sanju09 wrote:A right circular cone is inscribed in a hemisphere so that the base of the cone coincides with the base of the hemisphere. What is the ratio of the height of the cone to the radius of the hemisphere?
A. 2: 1
B. (1/2): 1
C. 1: 1
D. √2: 1
E. √3: 1

If you inscribe the cone into the hemisphere so that they have the same circular base, then the height of the cone will be the radius of the hemisphere. h=r. C is the correct answer.

dtweah wrote: If you inscribe the cone into the hemisphere so that they have the same circular base, then the height of the cone will be the radius of the hemisphere. h=r. C is the correct answer.

Is this a rule of thumb that should be remembered? How is the height related to the circular base of the cone? Sorry for asking a basic question, but I completely missed this concept.

I feel that there are two places where there can be confusion in your mind. so, I will try to explain both of them:
1. The word inscribed shape means that shape is completely contained inside and "fits snugly" inside another geometrical shape.

https://en.wikipedia.org/wiki/Inscribed

2. The concept of radius and height of the cone
Part 1:
Definition & concept of radius: To draw a sphere we choose a point in 3-D and call it center, then we choose a fixed distance and call it the radius. Sphere is then drawn by joining all the points at distance radius from the center.
So, we can deduce that the distance of all the points on the curved surface of the sphere and the center of the sphere is equal to the radius.

A hemisphere is a sphere split in half. so, the same rule must apply, thus for a hemisphere also, the distance of all the points on the curved surface and the center is equal to the radius.

Part 2:
Now, height of a right circular cone is: The distance of the top most point (vertex) from the center of the base. Agree?

Part 3:
The circular base of the cone coincides with the circular base of the hemisphere. Thus, if you focus on only the base in a 2-D world, the circles which forms the base of the hemisphere and the cone are the one and the same, thus their centers also coincide.

Part 4:
From all this we can deduce that the vertex of the cone lies on the curved surface of the hemisphere. Thus, height of the cone = radius of hemisphere as their circular base's centers coincide.

Hi kapsii,

Thanks for being patient with me You are right, I completely missed out on the "inscribed" word. If you see my earlier explanation I did not have the cone inscribed in the hemisphere, which is the reason why i did not answer the question correctly.

Got it!!!

inmate wrote:Got it!!!

many congratulations