RE Position of a piston? The distance between the head of a piston and the end of a 8t cylindrical chamber is given by ?x(t)= t+1c? ?, for ?t ? 0 s. The radius of the cylinder is 4 cm. a.? Find the volume of the cha ? mber for ? ? 0. b.? Find the rate of change of t? he? volu?me ?V?(?t) for ?t ? 0. c.? Graph the derivative of the volume function. On what intervals is the volume increasing? Decreasing?

Solution: Step 1: Given the distance between the head of piston and the end of cylinder is given by x(t)= 8tcm,for t 0.the radius of the denote it by r is given by r=4cm t+1 (a) The volume of the chamber for t 0 is given by Let the volume of the chamber be v(t) then 2 v(t)=r x =(4 )2 8t ( r=4cm,x(t)= 8tcm) t+1 t+1 = 1t+1t cubic cm Step 2: (b)the rate of change of the volume v’(t) for t 0 is given by v’(t) v’(t)= (v(t)) dt = ( 128t ) dt t+1d t =128 dt(t+1) (t+1) (t)(t) (t+1) =128 ( dt 2dt ) (t+1) =128 ( (t+1)(1)(t))1) (t+1) 1 =128 ( (t+1)) 128 = ((t+1)2 Step 3: The graph of the derivative of the volume function is given below