Problem 160 - Calculus Volume 3

Question

Suppose the sides of a rectangle are changing with respect to time. The first side is changing at a rate of 2 in./sec whereas the second side is changing at the rate of 4 \(\mathrm{in} / \mathrm{sec} .\) How fast is the diagonal of the rectangle changing when the first side measures 16 in. and the second side measures 20 in.? (Round answer to three decimal places.)

Step-by-Step Solution

Answer

The diagonal changes at approximately 4.375 inches per second.

Step 1: Understand the Problem

We are given a rectangle with sides changing over time. We need to find the rate at which the diagonal is changing, given the rates of change of the sides.

Step 2: Apply the Pythagorean Theorem

The diagonal of a rectangle can be found using the Pythagorean theorem: \[ d = \sqrt{x^2 + y^2} \]where \(d\) is the diagonal, and \(x\) and \(y\) are the lengths of the sides.

Step 3: Differentiate with Respect to Time

Differentiate the Pythagorean equation with respect to time \(t\) to find the rate of change of the diagonal:\[ \frac{dd}{dt} = \frac{1}{2\sqrt{x^2 + y^2}} (2x \frac{dx}{dt} + 2y \frac{dy}{dt}) \]Simplifying gives:\[ \frac{dd}{dt} = \frac{x \frac{dx}{dt} + y \frac{dy}{dt}}{\sqrt{x^2 + y^2}} \]

Step 4: Substitute Given Values

Substitute \(x = 16\), \(y = 20\), \(\frac{dx}{dt} = 2\), and \(\frac{dy}{dt} = 4\) into the differentiated equation:\[ \frac{dd}{dt} = \frac{16 imes 2 + 20 imes 4}{\sqrt{16^2 + 20^2}} \]

Step 5: Calculate the Diagonal Length

Calculate the diagonal's length using the current values:\[ \sqrt{16^2 + 20^2} = \sqrt{256 + 400} = \sqrt{656} \]

Step 6: Calculate the Rate of Change of the Diagonal

Substitute the diagonal length into the differentiated equation:\[ \frac{dd}{dt} = \frac{16 imes 2 + 20 imes 4}{\sqrt{656}} = \frac{32 + 80}{\sqrt{656}} = \frac{112}{25.6} \approx 4.375 \]

Step 7: Round the Answer

Round the rate of change of the diagonal to three decimal places: \[ \frac{dd}{dt} \approx 4.375 \]

Key Concepts

Pythagorean theorem

The Pythagorean theorem is a fundamental principle in geometry that connects the lengths of the three sides of a right triangle. It states that in a right triangle with sides of length \(x\), \(y\), and hypotenuse \(d\), the relationship is \(d^2 = x^2 + y^2\). This theorem is incredibly useful when dealing with problems involving rectangular shapes, such as a rectangle or square.

This concept is particularly handy for finding the diagonal of any rectangle. If you know the lengths of the rectangle’s sides, the Pythagorean theorem allows you to calculate the diagonal, which acts as the hypotenuse of a right triangle formed by the sides of the rectangle.

In the context of our problem, this is the crucial step needed to relate side lengths to the diagonal, allowing us to explore how changes in the sides affect the diagonal.

differentiation

Differentiation is a process in calculus used to determine the rate at which a quantity changes. It involves calculating the derivative, which represents the instantaneous rate of change of a function concerning one of its variables.

When we're dealing with time-dependent changes, differentiation allows us to compute how one variable changes as another variable changes with respect to time. In our problem, differentiation is utilized to find how the change in the sides of the rectangle affects its diagonal over time.

By differentiating the Pythagorean equation for the rectangle, we effectively examine how fast the diagonal changes. This involves understanding the relationship \( \frac{dd}{dt} = \frac{x \frac{dx}{dt} + y \frac{dy}{dt}}{\sqrt{x^2 + y^2}} \), which embodies how changes in \(x\) and \(y\) per time unit dictate the change in \(d\).

Practicing differentiation in this way helps in mastering the approach of tackling related rates problems, which is a common type of problem in calculus.

rate of change

The rate of change in mathematics is a measure of how a variable changes over a specified amount of time. In related rates problems, we're interested in the rate at which one variable changes as another variable changes.

For our example, we have two sides of a rectangle changing with time, and we need to determine the rate at which the diagonal between them is changing. The given rates—such as the side lengths changing at 2 inches per second and 4 inches per second—are integral to deducing how fast the diagonal changes.

Calculating the rate of change often involves identifying known rates (like \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \)) and applying them in the differentiated relation. Always keep in mind that every rate conveys information about speed or intensity, so understanding foundation rates helps piece together secondary rates, like the diagonal's rate of change.

rectangular geometry

Rectangular geometry deals with the properties and relations among all elements within rectangles. Rectangles have opposite sides that are equal and four right angles, making them fundamental in mathematics and practical applications.

Rectangular geometry is important when solving problems that involve related rates, mainly because of its simplicity and clear connection to the Pythagorean theorem. In problems like these, rectangles can transform our understanding by simplifying the spatial relationships between lateral dimensions and their diagonals.

In our problem, understanding the relationship among the rectangle’s sides and its diagonal is a key step. As sides of the rectangle change, the diagonal's length varies too. Such geometric insight gives intuition as to how various dimensions affect each other, essential for calculating related rates and more complex geometric problems.