Question

Can someone help me? Please.

A conical water tank with vertex down has a radius of 10 feet at the top and is 26 feet high. If water flows into the tank at a rate of 20 cubed feet per min, how fast is the depth of the water increasing when the water is 13 feet deep?

The depth of the water is increasing at _____ ft/min.

Answer 1

well from my reading of the question you need to find the rate of change in hieght with respect to time. So it looks like a related rate question

You know

\[\frac{dV}{dt} = 20 \]

you also know

\[V =\frac{1}{3} \pi r^2 h\]

and you know r = 10 and h = 26

I'd take the volume formula and make h the subject

\[h = \frac{3V}{\pi r^2}\]

then differentiate height with respect to volume

\[\frac{dh}{dV}\]

then you can say

\[\frac{dh}{dt} = \frac{dh}{dV} \times \frac{dV}{dt} \]

when you have \[\frac{dh}{dV}\] substitute r = 10 and then evaluate for \[\frac{dh}
{dt}\]

hope this helps

Answer 2

Could you evaluate your explanation a little more. I understand what you're saying but I'm coming out with weird number. @campbell_st

Answer 3

well what answer did you get...?

Answer 4

ummm I think I'm missing something

Answer 5

I got 26.74 haha. and it's wrong. @campbell_st

could you possibly lay out the problem, so I can see where I went wrong..

Answer 6

I got that too the first time, but it's wrong..

Answer 7

ummm its related rates by I can't get it

do you know the answer..?

Answer 8

I don't know tht answer. It's an online problem, it doesn't give hints. It just tells you if you're wrong or right. Which makes it even more complicated. haha @campbell_st

Answer 9

well you need to find the radius when the height is 13 ft..

the volume of the cone is

\[V = 2600 \pi\]

the height is now half the original

half the linear means 1/8 of the volume

so to find the radius

\[\frac{2600 \pi}{8} = \frac{1}{3} \pi r^2 \times 13\]

so \[r = \sqrt{\frac{3 \times 2600}{8 \times 13}}\]

so use the related rates shown early with the new r value.

Answer 10

which gives a rate of \[\frac{4}{5\pi} feet/\min\]

Answer 11

sorry I have nothing beyond that.

Answer 12

Ohhhhhhh! I totally missed the hlf a linear part! I got it now! Thank you so much! I really appreciate it (: @campbell_st

Answer 13

hope it helps.... and it does make a little sense in the the change in the height at 1/2 height is quicker than full height..

but good luck.