{Use of Tech} Flow from a tank A cylindrical tank is full at...

Question

{Use of Tech} Flow from a tank A cylindrical tank is full at time t=0 when a valve in the bottom of the tank is opened. By Torricelli’s law, the volume of water in the tank after t hours is V=100(200−t)², measured in cubic meters.
b. How long does it take for the tank to empty?

b. How long does it take for the tank to empty?

Accepted Answer

Hello. Today we're going to be solving the following word problem. So we are told that at a time t is equal to 0, the drain at the bottom of a cylindrical container full of oil is opened. The volume of the container after t amount of hours is given by the function v of t is equal to 50 multiplied by the quantity 250 minuteus t squared. This is where v is in cubic feet. We want to calculate the time that it will take for the container to be empty. So let's just go ahead and just draw a quick picture. So we have this container that is in the form of a cylinder. This cylinder has some type of liquid inside of it, and it has an opening at the bottom. So we're going to allow the variable t to equal the time. And at the time t is equal to 0, the container is opened and fluid starts to leak out. We want to use the volume function which is given to us as v is equal to 50 multiplied by 250 minuteus t squared and we want to use this to figure out how long it will take for this container to be completely empty. So what this means is that we want this container to have a volume of 0 or in other words, our target is when v of t is equal to 0. And our goal is to solve for how long it will take for this to equal to 0. So the first thing that we can do is we can take our volume function, 50 multiplied by 250 minuteus t squared and we're going to set it equal to 0. Then what we want to do is we want to solve for the variable t to see how long it will take for the cylinder to completely empty. So in order to solve for t, the first thing we can do is we can divide both sides of this equation by 50. Doing so will allow us to simplify the left hand side of this function as 250 minuteus t squared, and on the right hand side, 0 divided by 50 will leave us with the value of 0. We want to eliminate the exponent on the left hand side of the equation, so we're going to take the square root of both sides of the equation. Doing so will leave us with 250 minuteus t equal to the square root of 0, which is 0. The next thing we will do is we will subtract 250 from both sides of the equation leaving us with negative t is equal to negative 250 and finally, we're going to divide both sides of the equation by negative one and that will leave us with t is equal to positive 250. Now recall that t is in terms of hours, so that means that it is going to take us 250 hours in order for this container to have a volume of 0. And with that being said, the answer to this problem is going to be d. So I hope this video helps you in understanding how to approach this problem, and I'll go ahead and see you all in the next video.

{Use of Tech} Flow from a tank A cylindrical tank is full at time t=0 when a valve in the bottom of the tank is opened. By Torricelli’s law, the volume of water in the tank after t hours is V=100(200−t)², measured in cubic meters.
b. How long does it take for the tank to empty?

Key Concepts

Torricelli's Law

Volume Function

Finding Roots of a Function

Watch next

{Use of Tech} Flow from a tank A cylindrical tank ​​is full at time t=0 when a valve in the bottom of the tank is opened. By Torricelli’s law, the volume of water in the tank after t hours is V=100(200−t)², measured in cubic meters.b. How long does it take for the tank to empty?

Key Concepts

Torricelli's Law

Volume Function

Finding Roots of a Function

Watch next

{Use of Tech} Flow from a tank A cylindrical tank is full at time t=0 when a valve in the bottom of the tank is opened. By Torricelli’s law, the volume of water in the tank after t hours is V=100(200−t)², measured in cubic meters.
b. How long does it take for the tank to empty?