'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?'

my incorrect solution

where is my mistake?

  • 1
    $\begingroup$ why do you think your answer is wrong? $\endgroup$ – abel Jan 13 '15 at 0:32
  • $\begingroup$ Calculate the water area at that depth and multiply by 4 as a second check. $\endgroup$ – Joffan Jan 13 '15 at 0:33
  • $\begingroup$ 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... $\endgroup$ – Joffan Jan 13 '15 at 0:37
  • $\begingroup$ back of the book says the answer should be 25π/12 ft^3/min = 6.55 ft^3/min. $\endgroup$ – anatta Jan 13 '15 at 1:00
  • $\begingroup$ I have to second @Joffan with making the volume formula only in terms of $h$ very early on. $\endgroup$ – 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

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

  • $\begingroup$ back of the book says the answer should be 25π/12 ft^3/min = 6.55 ft^3/min. $\endgroup$ – anatta Jan 13 '15 at 1:07
  • $\begingroup$ @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. $\endgroup$ – abel Jan 13 '15 at 1:23
  • $\begingroup$ 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. $\endgroup$ – anatta Jan 13 '15 at 1:32
  • 1
    $\begingroup$ @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. $\endgroup$ – Joffan Jan 13 '15 at 1:36
  • $\begingroup$ @joffan good call. that does make a lot of sense. thanks. $\endgroup$ – anatta Jan 13 '15 at 1:44

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.