conical tank - radius of cone of water always porportional of height?

[Barry75]

Hi,

I'm taking a calculus class and working on a problem with the area of a conical tank from which water is draining.

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Water is draining out of a conical tank at a constant rate of 120 cubic inches per minute. How fast is the level of water in the tank decreasing when there are 7 inches of water in the tank?

(Hint: Notice that the radius of the cone will always be half the height, because the cone of water is always similar to the tank itself.)

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I know the volume of a conical tank is (1/3)(pi)(r²)(h) and I'm being told in the answer that I can replace r with h/2 in the formula as the cone of water in the tank is always similar to the tank itself. I've attached jpeg that shows the original question, the picture of the conical tank and the part of the answer that has me confused. I'm not sure if this is truly a geometry question but can someone please explain more details, if possible, why I can always assume the radius is half the height? I can see the cone of water and I can see the radius and height is decreasing...

Thanks,
Barry

 

[Deleted member 4993]

Hi,

I'm taking a calculus class and working on a problem with the area of a conical tank from which water is draining.

---

Water is draining out of a conical tank at a constant rate of 120 cubic inches per minute. How fast is the level of water in the tank decreasing when there are 7 inches of water in the tank?

(Hint: Notice that the radius of the cone will always be half the height, because the cone of water is always similar to the tank itself.)

---

I know the volume of a conical tank is (1/3)(pi)(r²)(h) and I'm being told in the answer that I can replace r with h/2 in the formula as the cone of water in the tank is always similar to the tank itself. I've attached jpeg that shows the original question, the picture of the conical tank and the part of the answer that has me confused. I'm not sure if this is truly a geometry question but can someone please explain more details, if possible, why I can always assume the radius is half the height? I can see the cone of water and I can see the radius and height is decreasing...

Thanks,
Barry

Do you the properties of similar triangles?

Do you know that if there are two similar triangles ABC and DEF, then:

AB/DE = BC/EF = CA/FD?

The partially filled cone can be represented by two similar triangles - the original cone and the part filled with fluid.

 

[Barry75]

Hi Subhotosh,

Sorry, I forgot about similar triangles since I took geometry MANY years ago! I just did more reading on the topic and it makes complete sense now.

Thanks,
Barry

P.S. Thanks stapel for fixing my post.