Please show work and answer all questions for points. 4) A tank in the form of a
ID: 2961143 • Letter: P
Question
Please show work and answer all questions for points.
4)
A tank in the form of a right circular cylinder of radius 2 ft and height 10ft is standing on one of its bases. The tank is initially full of water and water leaks from a circular hole of radius 1/2 inch at its bottom. First find the relationship between the rates the volume and height change with respect to time t. THen deteremine a differential equation for the height of the water at time t. Ignore friction and contraction of water at the hole and g=32ft/s^2.
5) For the autonomous first-order differential equation, y'=y^2+3y-18 find the critical points and phase portrait. CLassify the stability of each critical point.
6) Find the solution to the initial value problem, 2xy^2+(x^2-1)y'=0; y(0)=1.
7) Find the integrating factor for the differential equation and solve it, sec(x)y' +y-1=0.
Explanation / Answer
we have got equation in h and t. h is known at t = 0 and we can calculate t when h = 0.
dh/dt=-5/(6h^(2/3))
Or, h^(2/3)dh = (-5/6)dt
Integrating, we get (3/5)h^5/3 = (-5/6)t+k
at t = 0, h = 20ft; hence, (3/5)20^(5/3) = k
or, (3/5)h^(5/3) = (-5/6)t+(3/5)20^(5/3)
at h = 0, we have (-5/6)t+(3/5)20^(5/3) = 0
or, t = (18/25)*20^(5/3) = 106.1 secs. Ans.