Differential Equation (1 Viewer)

[skillz]

A conical tank has a radius length at the top equal to its height. Water, intially with a depth of 25cm, leaks out through a hole in the bottom of the tank at the rate of 5 root h cm ^3/min where the depth is h cm at time t minutes.

construct a differential equation expressing dh/dt as a function of h and solve it.



ta

 

[alcalder]

Well, the volume of a cone is

V = 1/3 πr²h

In this case r=h

So V = 1/3 πh³

dV/dh = πh²

dV/dt = 5√h cm³/min

dh/dt = dV/dt . dh/dV

= 5√h
(πh²)

When t=0, h=25 cm

then dh/dt = 1/(25π) at t=0

But what does it mean "solve it"?? Solve it for what, h? t? dh/dt?

 

[who_loves_maths]

But what does it mean "solve it"?? Solve it for what, h? t? dh/dt?

^ When you are required to solve a differential equation, it means you are to produce a function of the independent variables which satisfies the differential equation.

e.g. the solution to the differential equation y'(x) = 2x, is given by y = x² + C
where y is the dependent variable, and x the independent.

Solutions to DEs are always functions (this includes constant functions) or curves in finite dimensional space, and not dimensionless numbers.

In the context of skillz's question, you are being asked to formulate a differential equation involving the dh/dt term and to solve it - i.e. find the function h(t). [that is to say, how does h vary in relation to time?]