Comparative Rates of Change - Leaking Right Circular Cone

[Aminta_1900]

Hi,

This question pertains to comparative rates of change. The way I am working out the question below is similar to one of the examples given before the exercise begins in my textbook, so I may be taking it too literally.

Somewhere I have gone wrong, so I hope someone can kindly look at my workings out and find the discrepancy.

"A container in the form of a right circular cone of height 16 cm and base radius 4 cm is held vertex downward and filled with liquid. If the liquid leaks out from the vertex at a rate of 4 cm^3/s, find the rate of change of the depth of the liquid in the cone when half of the liquid has leaked out".

By "right circular cone" I take it to mean with an angle of 90deg at the vertex.

My workings....



Many thanks for your time.

Neil.

 

[skeeter]

Using similar triangles, r = h/4

[imath] V = \dfrac{\pi}{3} \cdot \dfrac{h^3}{16}[/imath]

 

[blamocur]

By "right circular cone" I take it to mean with an angle of 90deg at the vertex.

I don't see how you can have a 90 deg angle and r=h when the problem states h=16 and r=4.

 

[khansaheb]

By "right circular cone" I take it to mean with an angle of 90deg at the vertex.

First draw a sketch of a cone.

Right circular cone means the line joining the apex of the cone and the center of the base circle is normal (90°) to the plane of the base circle.

Thus the line indicating height joins he apex and the center of the base circle.

Now flip it such that the base circle is at the top and the apex is at the bottom → inverted cone.

 

[Steven G]

The radius is constant while the height of the water is decreasing, so how can you say that r= h*tan 45^o

 

[skeeter]

The radius is constant while the height of the water is decreasing, so how can you say that r= h*tan 45^o

the radius of the water level decreases as the water height in the cone decreases ...

[imath]\dfrac{h}{4} = r \implies \dfrac{dr}{dt} = \dfrac{1}{4} \cdot \dfrac{dh}{dt}[/imath]

 

[Harry_the_cat]

 

[Steven G]

View attachment 36433

I read cone and thought cylinder. You got me good!