Question

At the bottom of the tank containing 100 gallons of water, a leak was formed of diameter “D” cm that can empty 5 gallon/minute. Due to the load of water the diameter (D) of the leak started varying every minute with respect to (t) as “Dt” cm. In what time the whole tank will become empty?

Answer 1

Answer:

h

Step-by-step explanation:

Answer 2

The answer is 3.915 minutes.

Given:

The tank has 100 gallons of water

A leak of diameter D can empty 5 gallon/minute

DIameter changes every minute with Dt

To Find:

In what time the whole tank will become empty?

Solution:

Let the amount of water in the tank at a particular instant be x gallons.

The rate of emptying the tank is $\frac{dx}{dt}$.

The volumetric flow of the leak is proportional to the area of the leak.

Hence as given,

$5=k D^2$

as the area is proportional to the square of the diameter of the leak.

Let the time it takes to empty the tank be T.

Hence the volumetric output of diameter Dt is

$kD^2t^2=5t^2$

Hence we can write the rate as

$\frac{dx}{dt} = -5t^2$

Negative sign as the tank is emptying.

$dx=5t^2 .dt\\\int\limits^{0}_{100} \, dx =-\int\limits^T_0 {5t^2} \, dt$

$100=-\frac{5}{3}[T^3-0^3]\\ \\ 100= \frac{5}{3}[T^3]\\\\T=3.915$

Hence the time taken to empty the tank is 3.915 minutes.

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