Answer:
[tex]a)\ V=3053.628059 \ ft^3[/tex]
[tex]b)\ V'=1017.876\ ft^3/day[/tex]
Explanation:
Rate of Change
The volume of a cone of radius r and height h is given by
[tex]\displaystyle V=\frac{\pi r^2h}{3}[/tex]
The height is said to be 1/2 of the radius, thus
[tex]\displaystyle V=\frac{\pi r^2\cdot r}{2\cdot 3}[/tex]
[tex]\displaystyle V=\frac{\pi r^3}{6}[/tex]
a) Knowing r=18 feet, the volume is
[tex]\displaystyle V=\frac{\pi 18^3}{6}[/tex]
[tex]V=3053.628059 \ ft^3[/tex]
b) The rate of change of the volume is computed by taking the derivative of both sides respect to the time
[tex]\displaystyle V'=3\frac{\pi r^2}{6}r'[/tex]
[tex]\displaystyle V'=\frac{\pi r^2}{2}r'[/tex]
Where r' is the given rate of change of the radius: 2 feet/day.
Now we compute
[tex]\displaystyle V'=\frac{\pi 18^2}{2}\cdot 2=1017.876[/tex]
[tex]\boxed{V'=1017.876\ ft^3/day}[/tex]
