Respuesta :
Answer:
[tex]0.2122\text{ ft per min}[/tex]
Step-by-step explanation:
Let r be the radius ( in feet ) of the cone,
So, the diameter of the cone = 2 × radius = 2r feet,
∵ The diameter of the base of the cone is three times the height,
If h represents the height of the cone,
⇒ 2r = 3 × h,
⇒ [tex]r=\frac{3h}{2}[/tex] feet,
∵ Volume of a cone,
[tex]V=\frac{1}{3}\pi R^2 H[/tex]
Where,
R = radius, H = height,
Thus, the volume of the given cone is,
[tex]V=\frac{1}{3}\pi r^2 h[/tex]
[tex]=\frac{1}{3}\pi (\frac{3h}{2})^2 h[/tex]
[tex]=\frac{3}{4}\pi h^3[/tex]
Differentiating with respect to t ( time )
[tex]\frac{dV}{dt}=\frac{9}{4}\pi h^2\frac{dh}{dt}[/tex]
We have,
[tex]\frac{dV}{dt}=6\text{ cubic ft per min},h=2\text{ feet}[/tex]
[tex]6=\frac{9}{4}\pi (2)^2 \frac{dh}{dt}[/tex]
[tex]6=9\pi \frac{dh}{dt}[/tex]
[tex]\implies \frac{dh}{dt}=\frac{6}{9\pi }\approx 0.2122\text{ ft per min}[/tex]
