Answer:
The volume is changing at a rate given by:
[tex]\frac{dV}{dt} =19559.56\,\,\frac{in^3}{s}[/tex]
Step-by-step explanation:
Let's recall the formula for the volume of acone, since it is the rate of the cone changing what we need to answer:
Volume of cone = [tex]\frac{1}{3} B\,*\,H[/tex]
where B is the area of the base (a circle of radius R) which equals = [tex]\pi\,R^2[/tex]
and where H stands for the cone's height.
We apply the derivative over time operator ([tex]\frac{d}{dt}[/tex]) on both sides of the volume equation, making sure that we apply the rule for the derivative of a product:
[tex]V=\frac{1}{3} B\,*\,H\\\\V= \frac{1}{3} \pi\,R^2\,H\\\frac{dV}{dt} =\frac{\pi}{3}\,( \frac{d(R^2)}{dt} H+R^2\,\frac{dH}{dt} )\\\frac{dV}{dt} =\frac{\pi}{3}\,( 2\,R\,\frac{dR}{dt}\, H+R^2\,\frac{dH}{dt} )\\\frac{dV}{dt} =\frac{\pi}{3}\,( 2\,(110\,in)(1.4\,\frac{in}{s} )\,(151\,in)+(110\.in)^2\,(-2.3)\frac{in}{s} )\\\frac{dV}{dt} =19559.56\,\,\frac{in^3}{s}[/tex]
