Rate of Change in Water Level in a Conical Tank: A Calculus Problem

Understanding how the level of water changes in a conical tank over time is a fascinating application of calculus. This article will explore a specific problem involving a conical tank with water flowing into it at a steady rate. By the end, you will grasp the concepts of rate of change and how to apply calculus to solve such problems.

Introduction to Conical Tank Problem

A conical tank, standing point-down, has a height of 20 feet and a base radius of 5 feet. Water is being pumped into the tank at a rate of (8pi) cubic feet per minute. The question is, how fast is the radius of the surface of the water changing when the water is 4 feet deep?

The Volume of a Conical Tank

The volume (V) of a cone is given by the formula:

V13πr2h, where (r) is the radius and (h) is the height of the cone.

Relationship Between Radius and Height

Because the tank is a cone, the radius and height are proportional. The relationship can be expressed mathematically as:

v13πr2h

Dividing both sides by (frac{1}{3}pi), we get:

r3vπhfrac{1}{2}

Rate of Change of Radius

To find the rate of change of the radius, we need to differentiate the volume with respect to time and relate it to the rate of change of the radius.

V13πr2h

Differentiating both sides with respect to time (t), we get:

dV/dt23πrhdr/dt

Given that (frac{dV}{dt} 8pi), (h 4), and (r frac{5}{20}h frac{1}{4}h 1) (at (h 4)), we substitute these values:

8π23π1·4dr/dt

Solving for (frac{dr}{dt}):

dr/dt8π·32π41243·π/4

Thus, the radius of the surface of the water is changing at the rate of (3pi/4) feet per minute.

Conclusion

This problem demonstrates the practical application of calculus in real-world scenarios. By understanding the rate of change of volume and how it affects the radius of the tank, we can better predict and manage fluid levels in conical containers.

Key takeaways from this problem are:

The relationship between the volume, height, and radius in a cone. Using calculus to find the rate of change. Understanding the dynamics of fluid flow in conical tanks.

By breaking down the problem and applying calculus principles, we can confidently solve similar rate of change problems in various fields, including engineering and environmental science.

If you're interested in delving deeper into calculus and its applications, continue exploring problems like this to build your skills and knowledge.