Question

Rate of Change \(A\) 25-foot ladder is leaning against a house (see figure). If the base of the ladder is pulled away from the house at a rate of 2 feet per second, then the top will move down the wall at a rate of $$ r=\frac{2 x}{\sqrt{625-x^{2}}} \mathrm{ft} / \mathrm{sec} $$ where \(x\) is the distance between the base of the ladder and the house, and \(r\) is the rate in feet per second. (a) Find the rate \(r\) when \(x\) is 7 feet. (b) Find the rate \(r\) when \(x\) is 15 feet. (c) Find the limit of \(r\) as \(x\) approaches 25 from the left.

Short answer

The rate when \(x\) is 7 feet is \(r_a\). The rate when \(x\) is 15 feet is \(r_b\). The limit of the rate \(r\) as \(x\) approaches 25 from the left is negative infinity.

Step-by-step solution

Substitute the given values into the rate equation

For part (a), substitute \(x = 7\) into the rate equation \(r = \frac{2x}{\sqrt{625 - x^{2}}}\) to find \(r_a\). Similarly, for part (b), substitute \(x = 15\) into the rate equation to find \(r_b\).

Calculate the rates

For part (a), the rate comes out to be \(r_a = \frac{2*7}{\sqrt{625 - 7^{2}}}\) feet per second. Simplifying this gives \(r_a\). For part (b), the rate is \(r_b = \frac{2*15}{\sqrt{625 - 15^{2}}}\) feet per second, simplifying this gives \(r_b\).

Find the limit

For part (c), find the limit of the rate as \(x\) approaches 25 from the left. This is mathematically represented as \(\lim_{x\to 25^-} \frac{2x}{\sqrt{625 - x^{2}}}\). This limit calculation indicates what happens to the rate as the ladder nearly becomes perpendicular to the ground.

Evaluate the limit

Evaluate this limit using limit rules. It is seen that as \(x\to 25^-\), the denominator approaches zero from the left, while the numerator approaches 50, leading to the rate approaching negative infinity. This reflects the reality that as the horizontal distance \(x\) approaches the length of the ladder, the top of the ladder would slide down at an increasingly fast rate.

Key concepts

Limits in Calculus

In calculus, limits are essential for understanding how functions behave as they approach a specific point. Simply put, a limit looks at what happens to the value of a function as the input (usually denoted by 'x') gets infinitely close to some number. For instance, in the ladder problem, we are interested in the limit of the rate of change as the base of the ladder, denoted by 'x', approaches the length of the ladder, which is 25 feet.
To compute the limit, especially when it leads to an indeterminate form like 0/0 or infinity/infinity, we often employ algebraic simplification, L'Hôpital's rule, or special limit properties. In this ladder scenario, as 'x' approaches 25 from the left, the rate 'r' becomes extremely large in magnitude. This is because the denominator of our rate equation approaches 0, indicating that the ladder will move down the wall at a very high speed as it gets closer to the ground. This concept of limits is a cornerstone in calculus as they form the foundation for defining derivatives and integrals.

Related Rates

Related rates problems in calculus are a practical application of derivatives to understand how different quantities change together over time. In our exercise with the moving ladder, we are dealing with a related rates problem because the rate at which the base of the ladder moves relates to the rate at which the top goes down the wall.
To solve related rates problems, you typically follow these steps: identify the variables that are changing and relate them using an equation; differentiate this equation with respect to time to find a relationship between their rates of change; and finally, substitute known values to find the desired rate. By applying these steps to our ladder problem, we calculate the different rates 'r' at various distances 'x' to understand how quickly the top of the ladder descends as the base is pulled away.

Applying Derivatives

Derivatives represent the rate of change of a function with respect to a variable, and they are a central tool in solving rate of change problems in calculus. When you differentiate a function, you're finding a formula for the slope of the tangent line to the function's graph at any point. In real-world problems—like our ladder situation—you're often finding how fast some quantity is changing at any instant.
In context, the derivative of the ladder's position with respect to time gives us the rate at which the ladder is falling. By applying derivatives, we can predict the behavior of moving objects, optimize functions for maximum or minimum values, and analyze the rate at which quantities change. The ability to apply derivatives in such practical problems is what makes calculus such a powerful mathematical tool in various scientific and engineering fields.