Mathematics: Alandscaping company is pouring rock chips into a conical pile with a constant ratio of 2: 5 between the radius and height.the volume of the rock chips is increasing at a rate of 1.8m3/min.at what rate is the height increasing when the radius is 3

The probability = 7p4 / 14p4...

Alandscaping company is pouring rock chips into a

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campbelldean

02.01.2022

$\bf \textit{volume of a cone}\\\\
V=\cfrac{\pi r^2 h}{3}\impliedby \textit{the r:h ratio is 2:5}\implies \cfrac{r}{h}=\cfrac{2}{5}\implies r=\cfrac{2h}{5}
\\\\\\
V=\cfrac{\pi \left(\frac{2h}{5} \right)^2 h}{3}\implies V=\cfrac{\frac{\pi 2^2h^2h}{5^2}}{3}\implies V=\cfrac{4h^3\pi }{25}\cdot \cfrac{1}{3}
\\\\\\
V=\cfrac{4\pi h^3}{75}\\\\
-------------------------------\\\\$

$\bf \cfrac{dv}{dt}=\cfrac{4\pi }{75}\cdot 3h^2\cfrac{dh}{dt}\implies \cfrac{dv}{dt}=\cfrac{4\pi h^2 }{25}\cdot \cfrac{dh}{dt}\implies \cfrac{25\frac{dv}{dt}}{4\pi h^2}=\cfrac{dh}{dt} 
\\\\\\
\begin{cases}
\frac{dv}{dt}=1.8\\
r=3\\
h=\frac{5}{2}r\\
h=\frac{15}{2}
\end{cases}\implies \cfrac{25\cdot 1.8}{4\pi \left( \frac{15}{2} \right)^2}=\cfrac{dh}{dt}$

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Alandscaping company is pouring rock chips into a conical pile with a constant ratio of 2: 5 between the radius and height.the volume of the rock chips is increasing at a rate of 1.8m3/min.at what rate is the height increasing when the radius is 3

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