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STD 12 - 6. Application of derivatives — Mathematics STD 12 Science — Question

CBSE BoardEnglish MediumSTD 12 ScienceMathematicsSTD 12 - 6. Application of derivatives1 Mark
MCQ
Coffee is draining from a conical filter, height and diameter both $15 \,\,cms$ into a cylinderical coffee pot diameter $15 \,\,cm$. The rate at which coffee drains from the filter into the pot is $100 \,\,cu cm /min$.The rate in $cm s/min$ at which the level in the pot is rising at the instant when the coffee in the pot is $10 \,\,cm$, is
  • A
    $\frac{9}{16\pi }$
  • B
    $\frac{25}{9\pi }$
  • C
    $\frac{5}{3\pi }$
  • ✓
    $\frac{16}{9\pi }$
✓

Answer

Correct option: D.
$\frac{16}{9\pi }$
d
For cylindrical pot  $V = \pi r^2h$

$\frac{{dV}}{{dt}} = \pi \left[ {{r^2}\frac{{dh}}{{dt}} + h\,\cdot\,2r\frac{{dr}}{{dt}}} \right]$(r = constant, $\frac{{dr}}{{dt}} = 0$)

hence,$100 = \pi r^2\frac{{dh}}{{dt}}$

$100 = \pi · \frac{{225}}{4} · \frac{{dh}}{{dt}}$(r = $\frac{{15}}{{2}} cm$)

$\frac{{dh}}{{dt}} = \frac{{400}}{{225\pi }} = \frac{{16}}{{9\pi }} cm/min$

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