4.       A conveyor belt is dumping corn into a large empty silo, forming a circular based cone shaped - heap on the floor. The base radius is twice the size of the height of the pile. If the corn is filling in the space at a rate of 2 `*`(`^`(m, 3)); / minute, find the rate at which the height is changing when the height is 3 m deep. 

restart;  

 

 

algsubs(r = `+`(`*`(2, `*`(h))), `+`(`*`(`/`(1, 3), `*`(Pi, `*`(`^`(r, 2), `*`(h))))));  

`+`(`*`(`/`(4, 3), `*`(Pi, `*`(`^`(h, 3))))); (8.4.1)
 

V := proc (t) options operator, arrow, function_assign; `+`(`*`(`/`(4, 3), `*`(Pi, `*`(`^`(h(t), 3))))) end proc;  

V := proc (t) options operator, arrow; `+`(`*`(`/`(4, 3), `*`(Pi, `*`(`^`(h(t), 3))))) end proc; (8.4.2)
 

diff(V(t), t);  

`+`(`*`(4, `*`(Pi, `*`(`^`(h(t), 2), `*`(diff(h(t), t)))))); (8.4.3)
 

 

Vprime(t); (8.4.4)
 

subs(diff(h(t), t) = K, Vprime(t));  

`+`(`*`(4, `*`(Pi, `*`(`^`(h(t), 2), `*`(K))))); (8.4.5)
 

 

Vprime(t); (8.4.6)
 

VprimeGiven := 2;  

VprimeGiven := 2; (8.4.7)
 

h := proc (t) options operator, arrow, function_assign; 3 end proc;  

h := proc (t) options operator, arrow; 3 end proc; (8.4.8)
 

solve(VprimeGiven = Vprime(t), K);  

`+`(`/`(`*`(`/`(1, 18)), `*`(Pi))); (8.4.9)