Problem 77 - Single Variable Calculus

Question

A cylindrical tank of height \(h\) is filled with water. Suppose jet of water flows through an orifice on the tank. According to Torricelli's law, the velocity of flow of the jet of water is given by \(V=\sqrt{2 g x}\) where \(g\) is the gravitational constant. It can be shown that the range \(R\) (in feet) of the jet of water is given by \(R=2 \sqrt{x(h-x)}\). Where should the orifice be located so that the jet of water will have the maximum range?

Step-by-Step Solution

Answer

The orifice should be located at the midpoint (height \(h/2\)) of the cylindrical tank to achieve maximum range.

Step 1: Calculate the derivative of the range function

Firstly, let's find the derivative of the range function, R(x) = 2√(x(h - x)). This will help us to find the critical points where the maximum range can be observed. The range function can be rewritten as R(x) = 2(x(h - x))^(1/2). Now we can use the chain rule to differentiate R with respect to x (dR/dx): d[R(x)]/dx = (1/2) * 2 * (x(h - x))^(-1/2) * d[x(h - x)]/dx Now, we need to differentiate the inner function x(h - x) with respect to x: d[x(h - x)]/dx = h - 2x Then, substitute this back into the derivative of R: d[R(x)]/dx = (h - 2x)(x(h - x))^(-1/2)

Step 2: Find the critical points

Now, we have to find the critical points (where dR/dx = 0): (h - 2x)(x(h - x))^(-1/2) = 0 The derivative is equal to 0 when h - 2x = 0, so: 2x = h x = h/2

Step 3: Find the maximum range

We have found the critical point x = h/2, now we need to find the actual maximum range using the original range function: R(h/2) = 2 √((h/2)(h - h/2)) R(h/2) = 2 √((h/2)(h/2)) R(h/2) = 2√((h^2)/4) The maximum range of the jet of water is 2√((h^2)/4). So the orifice should be located at the midpoint (height h/2) of the cylindrical tank to achieve maximum range.

Key Concepts

Calculus in Physics

Calculus plays a vital role in physics, providing the tools to model and analyze motion, forces, and energy. The application of calculus allows for the precise prediction of how physical systems evolve over time. In the context of Torricelli's law problem given, calculus is utilized to determine the velocity of a jet of water flowing from a tank and to optimize the range of the projectile, which is the horizontal distance it covers.

Understanding the relationship between a function and its derivative is key in solving many physics problems. Here, the function used to calculate the range of a projectile is a result of physical laws applied to a real-world scenario. By finding the derivative of the range function, we can identify critical points which indicate a maximum or minimum value. This process aligns perfectly with the principles of calculus and allows us to interpret the physical implications of mathematical solutions.

Optimization Problems

Optimization problems are fundamental in both mathematics and physics. They involve finding the best solution from all feasible solutions, often by identifying the maximum or minimum values of a function. In our exercise, the optimization problem is to determine where the orifice in a cylindrical tank should be located to maximize the range of a jet of water.

To solve such problems, a common strategy is to use calculus to find the derivative of the function being optimized, in this case, the range. By setting the derivative equal to zero, we find the critical points that could potentially yield the maximum value. Additional tests, like the second derivative test, could be applied to confirm that these critical points are indeed maxima. Our step-by-step solution applies calculus to an optimization problem, showcasing how mathematical theory directly informs practical decisions.

Maximum Range of Projectile

The maximum range of a projectile is a concept of interest in many physics applications, from engineering to sports science. It refers to the furthest horizontal distance traveled by an object moving under the influence of gravity. In our exercise, we use a specific case of projectile motion governed by Torricelli's law.

This law relates the velocity of the water jet to the height of the water above the orifice, which in turn determines the range of the projectile. By finding where this range is maximized, we're addressing a classical question in projectile motion. The conditions for maximum range often involve the launch angle (in typical projectile problems) or, in our tank problem, the optimal location of the orifice. Hence, understanding how to calculate the maximum range is not only key for academic purposes but also has practical implications in various fields where projectile motion is relevant.

Derivative of Range Function

The derivative of the range function is the cornerstone for solving Torricelli's law problem. The range function, which gives the horizontal distance covered by the water jet, is a square root function that involves the initial height of water and the height above the orifice. To maximize this range, we take the derivative of the range function with respect to the variable of interest, which in this case is the height above the orifice.

The derivative provides the rate of change of the range with respect to this height, and when set to zero, indicates potential maximum or minimum points. We utilize calculus tools such as the chain rule for differentiation to calculate this derivative. In this approach, the complexity of the range function's formula is systematically broken down into simpler parts that are easier to differentiate. The steps provided in the solution illustrate this process and reveal the calculus concept in action, specifically how derivatives pave the way towards optimizing functions in physics problems.