# Related Rates - Water Flows into a Cylindrical Tank ## Метаданные - **Канал:** The Organic Chemistry Tutor - **YouTube:** https://www.youtube.com/watch?v=wLzl8qyOW-w ## Содержание ### [0:00](https://www.youtube.com/watch?v=wLzl8qyOW-w) Segment 1 (00:00 - 04:00) Number 99. Water flows into a cylindrical tank at a rate of 40 cubic feet per minute. How fast is the height of the water level changing when the radius of the cylinder is 6 ft? So, let's write down what we know. 40 cubic feet per minute. What does that represent? Cubic feet is a unit of volume. Volume is cubic feet, cubic inches, cubic yards. So, that's the volume flow rate. That's going to be dvdt. That's the rate at which water flows into the cylindrical tank. And we know that the radius of the cylinder is 6 ft. We want to find how fast the height of the water level is changing. So we look for dhct. Now let's draw a picture. So, let's say this is the cylinder and let's say there's water inside the cylinder and we're adding water into it. As we add water to the cylinder, the height of the water level in this cylinder is going to increase. So, here is H. And as we add water, H is going to go up. So, dht should be positive. But now notice the radius of the water in this cylinder. It's not going to change as we add water to it. The radius is going to stay constant because the shape of the cylinder remains constant. As a result, ddt is zero. R is treated as a constant. Now because we have the volume flow rate, we need to start with the volume for the volume formula of a cylinder and that's pi r^ 2 * h. So let's go ahead and find the derivative with respect to t. The derivative of v is 1 * dv dt. r 2 is a constant. So we can just rewrite p r^ 2. The derivative of h is 1 * dhdt. So this would be equivalent to finding the derivative of 8x. That will be 8. So the derivative of p r^ 2 * h is just p r^ 2, which is what we have. The h goes to one. But we do have to add the htt since we're differentiating it with respect to time. So now let's get dhdt by itself. Let's divide both sides by pi r 2. So we have dhdt is equal to dv dt and instead of dividing that by p r^ 2 we can write it as 1 / p r 2. This setup will help you to see what the units of DHCT should be. DVDT, this is 40 cubic feet per minute. And then 1 / r is 6 ft and we're going to square that. So 40 / 6 2 that's 10 over 9. If you divide that by pi, we get that dvdt is approximately this is a rounded answer 354. And when we divide cubic feet by square feet, here you have feet* feet. And here you only have feet* feet. Two of them will cancel, leaving one behind. So if you want to visually see it, this is feet time feet. And here this is just feet time feet. These two will cancel. We have one left over. And we have minutes on the bottom. So it's going to be feet per minute. And correction, this should be an H, not a V. So that's dhdt. So that's how fast the height of the water level is changing. It's increasing by almost a little bit more than a third of a foot every minute. --- *Источник: https://ekstraktznaniy.ru/video/41226*