Question 241152
A cylindrical container has 3 inches of water in it and is being filled at a rate of 1/2 inch per minute.
 The volume, V, of the water in the container is given by the function: V(h)=(π/4)h^3. 
Write a formula for the volume in terms of the time in minutes.
:
The formula appears to be the volume of a cylinder with a radius of 1/2 inch
V(h) = {{{pi*(1/2)^2 * h}}}
which is
V(h) = {{{pi*(1/4) * h}}}
or
V(h) = {{{(pi/4)*h}}}
:
let t = time in minutes, h = 3 inches to start with, therefore the height:
(3+.5t)
V(t) = {{{(pi/4)*(3+.5t)}}}
:
 Then, calculate the amount of time that will take to fill the container to 70 cubic inches.
Replace V(t) with 70, solve for t
{{{(pi/4)*(3+.5t)}}} = 70
Multiply both sides by {{{4/pi}}} (reciprocal gets rid of {{{(pi/4)}}} on the left)
3 + .5t = 70 * {{{4/pi}}}
:
3 + .5t = {{{280/pi}}}
3 + .5t = 89.126
.5t = 89.126 - 3
.5t = 86.126
t = {{{86.126/.5}}}
t = 172.25 min to reach 70 cu/inches
;
:
Check this in the volume equation: t=172.25
V(t) = {{{(pi/4)*(3+.5t)}}}
V(t) = {{{(pi/4)*(3+.5(172.25))}}}
V(t) = {{{(pi/4)*89.125}}}
V(t) = 69.9986 ~ 70
;
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Two hrs is a long time, hope this has helped!