# rate of change, water filling tank

'A tank is in the form of a cone with the point downward, and the height and diameter are each 10 feet. How fast is the water pouring in at the moment when it is 5 feet deep and the surface is rising at the rate of 4 feet per minute?'

where is my mistake?

• why do you think your answer is wrong? – abel Jan 13 '15 at 0:32
• Calculate the water area at that depth and multiply by 4 as a second check. – Joffan Jan 13 '15 at 0:33
• I should also say that you made some work for yourself (and increased the chance of error) by leaving the substitution $r=h/2$ so late... – Joffan Jan 13 '15 at 0:37
• back of the book says the answer should be 25π/12 ft^3/min = 6.55 ft^3/min. – anatta Jan 13 '15 at 1:00
• I have to second @Joffan with making the volume formula only in terms of $h$ very early on. – turkeyhundt Jan 13 '15 at 1:08

Your answer is right. here is a way you can avoid lots of work and volume formula and all that. At a particular instant, what is important is the surface area. You can forget about that it is cone: it might as well be a cylinder at that instant.

• Let $r$ be the radius at height $h$.
• From the geometry, $r = h/2$.
• You want to find ${dV \over dt}$ when $h = 5$ and $\frac{dh}{dt} = 4$.

The change in volume at this instant is

$$dV = \pi *(5/2)^2 dh$$

If you divide both sides by $dt$ you get

\begin {align} {dV \over dt} &= \pi * {25 \over 4} *4 \\ &= 25 \pi\ ft^3/min \\ \end{align}

• back of the book says the answer should be 25π/12 ft^3/min = 6.55 ft^3/min. – anatta Jan 13 '15 at 1:07
• @anatta, i think the answer in the back of the book is wrong if the cone has diameter and height are 10ft as we took. cheek the problem statement again. – abel Jan 13 '15 at 1:23
• i copied and pasted the problem. its from a pdf of 'calculus for the practical man'. was just curious the solutions section would tell me that my answer is off by a factor of 12. thank you for taking the time to review my work. – anatta Jan 13 '15 at 1:32
• @anatta I really suggest that the problem originally said "4 inches per minute". This fits the book answer exactly. Now who made the transcription error, I have no idea. – Joffan Jan 13 '15 at 1:36
• @joffan good call. that does make a lot of sense. thanks. – anatta Jan 13 '15 at 1:44