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If a tank holds 5000 gallons of water, which drains from the bottom of the tank in 40 minutes, then Torricelli's Law gives the volume $ V $ of water remaining in the tank after $ t $ minutes as

$ V = 5000 (1 - \frac {1}{40}t)^2 \space \space \space \space 0 \le t \le 40 $

Find the rate at which water is draining from the tank after (a) 5 min, (b) 10 min, (c) 20 min, and (d) 40 min. At what time is the water flowing out the fastest? The slowest? Summarize your findings.

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00:59

Amrita Bhasin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 7

Rates of Change in the Natural and Social Sciences

Derivatives

Differentiation

Baylor University

University of Michigan - Ann Arbor

Idaho State University

Boston College

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

03:24

If a tank holds 5000 gallo…

05:13

0:00

If a cylindrical water tan…

03:10

Torricelli's Law A ta…

04:18

A tank holds 50 gal of wat…

07:40

Okay, So this problem involves tourists Eli's Law, which describes the volume of water remaining in the tank as a function of time. So we have this volume equation, and we're interested in finding the rate of change, the rate at which water is draining at various times. So we need the derivative of the function. So we'll use the chain rule first will leave the constant 5000 will bring down the two and will raise the inside to the first. And then we'll multiply by the derivative of the insight. Now we can multiply our Constance together 5000 times, two times negative 1/40 and we end up with the prime of tea equals negative 2 50 times, one minus 1/40 t. So we're going to use that function. And for part A, we're going to find the prime of five. So a substitute five in there when we get negative to 18.75 and the units would be gallons per minute. And for part B, we find the rate of change at 10 minutes. So the prime of 10 we substitute 10 into the equation and we get negative 1 87.5 Callens per minute. It's negative because of volume is decreasing for part C for 20 minutes, the prime of 20. And that gives us negative 1 25 gallons per minute Notice the rate is changing As the tank empties, the rate is slowing. And then part D. V, prime of 40 well, noticed that it only takes 40 minutes to empty the tank. So, do you have a guess as to what this is going to be when we substitute 40 into the equation? We end up with zero. The tank is empty. Okay, so what do we noticed? When is it the fastest? And when is it this lowest? It's the fastest at the beginning. It slows down as it goes along and as a tank empties slowest at the end.

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