Problem solving Conical glass

[skiing4free]

This is a problem that my lecturer gave us in class and it has been bugging me ever since. I have been unsuccesful in finding or calculating a proper solution so I am hoping PF will be able to help...

This is the Q:

Let H be the height of a conical glass which is filled to a height h. Find the volume of the liquid in the glass as a proportion of the volume if the glass is full. Find the ratio h/H for which the glass is half full.
To answer this question you must construct a mathematical model defining all the variables.

This does not sound like a difficult question to solve but whenever I try to solve with simple conical volume equations I just get that the ratio is H:2h which is obviously not right. Any help would be greatly appreciated.

 

[Office_Shredder]

Why don't you explain how you got H:2h?

 

[skiing4free]

Well the formula for a volume of a cone is:
V=1/3*pi*r^2*H.

V1 which is the volume of the full cone and V2 is the volume of the half full cone it is clear that V1/2=V2. I put the volume equations into this with the two differnt heights and when the constants are removed (pi, 1/3) you are left with H=2h. ahh but i have just seen how the radius would of course change... Now I think I am more confused

 

[Office_Shredder]

Can you find a relationship between what the radius and the height is going to be when you're at a certain height up the cone? (hint: think similar triangles)

 

[CellCoree]

I understand the frustration that comes with being unable to solve a problem, especially one that has been assigned by a lecturer. Let's break down the problem and approach it systematically.

First, we need to define all the variables in the question. H is the height of the conical glass, h is the height of the liquid in the glass, and V is the volume of the liquid in the glass. We also know that the glass is filled to a height h, which means the remaining height of the glass is H-h.

Next, we need to construct a mathematical model to represent the volume of the liquid in the glass. We can use the formula for the volume of a cone, which is V = (1/3)πr^2h, where r is the radius of the base of the cone. However, we need to modify this formula to account for the fact that the glass is not full to the top. We can do this by multiplying the formula by the ratio of the height of the liquid to the total height of the glass (h/H), giving us the following equation: V = (1/3)πr^2h(h/H).

To find the volume of the liquid as a proportion of the volume if the glass is full, we need to divide the volume of the liquid by the volume if the glass was full, which can be represented as (1/3)πr^2H. This gives us the following equation: V/(1/3)πr^2H = (1/3)πr^2h(h/H) / (1/3)πr^2H. Simplifying this, we get V/(1/3)πr^2H = h/H.

To find the ratio h/H for which the glass is half full, we can set the volume of the liquid equal to half of the volume if the glass was full, which can be represented as (1/2)(1/3)πr^2H. This gives us the following equation: (1/2)(1/3)πr^2H = (1/3)πr^2h(h/H). Simplifying this, we get h/H = 1/2, meaning that the ratio h/H for which the glass is half full is 1:2.

I hope this helps you solve the problem and alleviate your frustration. Remember, when faced with