Calculus

Application of Derivatives

Medium

applied

Sand is pouring from a pipe at the rate of 12 $cm^3$ /sec. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm ?

$\frac{1}{80\pi}$ cm/sec

$\frac{1}{48\pi}$ cm/sec

$\frac{1}{36\pi}$ cm/sec

$\frac{1}{144\pi}$ cm/sec

✅ Correct Option: 2

Since $h=\frac{r}{6}$ , we have $r=6h$ and $V=\frac{1}{3}\pi(6h)^2h=12\pi h^3$ .

Then $\frac{dV}{dt}=36\pi h^2\frac{dh}{dt}=12$ , so at $h=4$ , $\frac{dh}{dt}=\frac{12}{36\pi(16)}=\frac{1}{48\pi}$ cm/sec.

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For which of the following real functions, $x=0$ is a point of local maxima?

(A) $f(x)=4x^4-3x^3-2x^2+1$

(B) $f(x)=3x^5+\sqrt{2}x^4-7x^2+4$

(D) $f(x)=-3x^3+2x^2-9$

Choose the correct answer from the options given below:

Sand is pouring from a pipe at the rate of 12 cm3cm^3cm3/sec. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm ?