Final answer:
The problem is solved by setting up a balance equation between the volume of water entering the tank, leaving the tank, and the initial volume, leading to a quadratic equation that can be solved to find the emptying time.
Step-by-step explanation:
The question is about determining the time it will take for a water tank to empty given the rates of water entering and exiting the tank. To find the time when the tank will be empty, we can set up the equation that equates the volume of the water leaving the tank to the initial volume plus the volume of the water entering.
Let's calculate the volume of water entering the tank every minute. The rate of water entering is 20 liters per minute, which is 0.02 m³/min (since 1 liter = 0.001 m³).
For the water leaving, the rate starts at 0 liters per minute and increases by 1 liter per minute every minute, thus it can be represented as t liters per minute where t is the time in minutes.
The initial volume of water in the tank is the volume of a cube with side 5 m, which is 5³ = 125 m³ (since the tank starts out filled to a depth of 5 meters).
Setting up the equation to represent balance between inflow and outflow:
- Total volume entered = 0.02t
- Total volume exited = (1/2)*t(t+1) (because the sum of the first t integers is (1/2)*t(t+1))
- Initial volume = 125 m³
Equating the initial volume and the volume exited minus the volume entered gives us:
125 = (1/2)*t(t+1) - 0.02t
This is a quadratic equation in terms of t which can be solved to find the time when the tank will be empty.