Question

a cylindrical tank with axis horizontal has a diameter d and a length l. it holds oil to depth d (d < d) when a leaks starts to drain the oil at a constant rate k. how fast is the level falling? (solve in calculus)

Answer 1

The rate at which the oil level is falling can be determined using calculus and the principles of fluid dynamics. We can use the formula V = πr^2h to calculate the volume of oil being drained per unit time. By taking the derivative of this formula, we can find the rate at which the oil level is falling.

Step-by-step explanation:

The rate at which the oil level is falling can be determined using calculus and the principles of fluid dynamics. We can start by considering the volume of oil being drained per unit time. Since the tank is cylindrical, the volume of oil drained can be represented by the formula V = πr^2h, where r is the radius of the tank and h is the height of the oil.

The rate at which the volume of oil is changing can be determined by taking the derivative of this formula with respect to time, t. Thus, the rate of change of volume is dV/dt = πr^2(dh/dt). We can solve for dh/dt by rearranging the equation as dh/dt = (dV/dt)/(πr^2).

Since h is decreasing, dh/dt will be negative. This represents the rate at which the oil level is falling.