Respuesta :
Answer:
[tex]\frac{dh}{dt}=0.071[/tex] m/min
Explanation:
You have to use the volume of a cone, which is:
[tex]V=\frac{1}{3}\pi r^{2}h[/tex]
where r is the radius of the base and h is the height.
In this case, r=5 and h=10. The radius can be written as r=h/2
Replacing it in the equation:
[tex]V=\frac{1}{3}\pi (\frac{h}{2})^{2} h=\frac{1}{12}\pi h^{3}[/tex] (I)
The rate of the volume is the derivate of volume respect time, therefore you have to perform the implicit differentiation of the previous equation and equal the result to 3.14 m³/min
[tex]\frac{dV}{dt}=\frac{\pi }{12}(3)h^{2}\frac{dh}{dt} =\frac{\pi }{4}h^{2}\frac{dh}{dt}[/tex]
Replacing dV/dt= 3.14, h=7.5 and solving for dh/dt, which represents how fast the level is rising:
[tex]3.14=\frac{\pi }{4}(7.5)^{2}\frac{dh}{dt}\\3.14=\frac{225\pi }{16}\frac{dh}{dt}[/tex]
Multiplying by 16/225π both sides:
[tex]\frac{dh}{dt}=0.071[/tex] m/min
