Differentiation

[pylrt]

According to Torricelli's law the time rate of change of the volume V of a water draining tank is proportional to the square root of the water;s depth. A cylindrical tank of radius [latex]10 \sqrt[]{\pi}[/latex] centimeters and height 16 centimeters, which was full initially, took 40 seconds to drain.

(a) Write a differential equation for V at time t and the two corresponding conditions.

(b)Solve the differential equation

(c)Find the volume of water after 10 seconds

 

[BigGlenntheHeavy]

\(\displaystyle Torricelli's \ Law:\)

\(\displaystyle a) \ \frac{dV}{dt} \ = \ k\sqrt h\)

\(\displaystyle Given: \ r \ = \ \frac{10}{\sqrt \pi}cm, \ h \ = \ 16cm, \ hence \ Volume \ of \ full \ tank \ = \ 1600cm^{3}.\)

\(\displaystyle dV \ = \ 4k dt, \ \int dV \ = \ \int(4k)dt, \ V(t) \ = \ 4kt+C\)

\(\displaystyle V(0) \ = \ 1600 \ =0+ \ C, \ C \ = \ 1600\)

\(\displaystyle Hence, \ V(t) \ = \ 4kt+1600, \ V(40) \ = \ 0 \ = \ 160k+1600, \ k \ = \ -10\)

\(\displaystyle b) \ Therefore \ V(t) \ = \ -40t+1600\)

\(\displaystyle and \ c), \ V(10) \ = \ 1200cm^{3}\)