Question

Changing Dimensions in a Rectangle The length \(\ell\) of a rectangle is decreasing at the rate of 2 \(\mathrm{cm} / \mathrm{sec}\) while the width \(w\) is increasing at the rate of 2 \(\mathrm{cm} / \mathrm{sec} .\) When \(\ell=12 \mathrm{cm}\) and \(w=5 \mathrm{cm},\) find the rates of change of. (a) the area, (b) the perimeter, and (c) the length of a diagonal of the rectangle. (d) Writing to Learn Which of these quantities are decreasing, and which are increasing? Explain.

Short answer

The rates of change when \(\ell = 12cm\) and \(w = 5cm\) are: (a) rate of change of area is \(\frac{dA}{dt}\), (b) rate of change of perimeter is \(\frac{dP}{dt}\), and (c) rate of change of the diagonal is \(\frac{dD}{dt}\). For (d), if a rate of change is positive, the quantity is increasing; if it is negative, the quantity is decreasing.

Step-by-step solution

Derive the equations for area, perimeter, and diagonal

Start by deriving the standard equations for the Area \(A\), Perimeter \(P\), and Diagonal \(D\) in terms of length \(\ell\) and width \(w\). Remember that: 1. Area \(A\) of a rectangle can be found by multiplying its length \(\ell\) and width \(w\). So, \(A = \ell w\). 2. Perimeter \(P\) is the sum of all sides of the rectangle, which gives \(P=2(\ell+w)\). 3. Diagonal \(D\) can be found using Pythagoras' theorem \(D=\sqrt{\ell^2 + w^2}\).

Calculate the rate of change of Area

To find the rate at which the area changes, take the derivative of the area equation with respect to time \(t\). This gives \(\frac{dA}{dt} = \frac{d}{dt} (\ell w) = \ell \frac{dw}{dt} + w \frac{d\ell}{dt}\).You know the values of \(\frac{dw}{dt}\) and \(\frac{d\ell}{dt}\) from the problem, and also the values of \( \ell \) and \( w \). Plug these into the equation to calculate \(\frac{dA}{dt}\).

Calculate the rate of change of Perimeter

Again, take the derivative of the perimeter equation with respect to time \(t\). This gives us:\(\frac{dP}{dt} = \frac{d}{dt}(2(\ell+w)) = 2 (\frac{d\ell}{dt} + \frac{dw}{dt})\). Plug in the known values to obtain \(\frac{dP}{dt}\).

Calculate the rate of change of Diagonal

For the diagonal, you take the derivative of the diagonal equation with respect to time \(t\). This gives:\(\frac{dD}{dt} = \frac{1}{2\sqrt{\ell^2 + w^2}} (2\ell \frac{d\ell}{dt} + 2w \frac{dw}{dt})\). Substitute the known values to compute \(\frac{dD}{dt}\).

Analyzing which quantities are increasing or decreasing

Analyze the signs of the rates of change \(\frac{dA}{dt}\), \(\frac{dP}{dt}\), and \(\frac{dD}{dt}\). A positive rate indicates an increasing quantity, while a negative rate indicates a decreasing quantity.

Key concepts

Derivative Applications

The concept of derivatives is fundamental to calculus and serves as a powerful tool for analyzing the way quantities change. In real-world scenarios, understanding how variables such as speed, acceleration, volume, and others change over time is essential for predictions and optimizations.

One common application is the rate of change. By taking the derivative of a function, you can determine how quickly or slowly a particular quantity is changing at a specific instance. This is seen in motion analysis where derivatives help in finding velocity and acceleration from a position-time graph. Within geometry, derivatives help calculate changing areas or volumes, which is precisely what is being explored in the exercise of a rectangle with changing dimensions.

Using the given rates of change for the length and width of a rectangle, and applying derivatives, we are able to compute how the area, perimeter, and diagonal length of the rectangle change over time, which has direct implications in fields like engineering and physics where dimensions may affect structural integrity or performance.

Related Rates Problems

Related rates are a class of problems in calculus that involve finding the rate at which one quantity changes with respect to another, when the quantities are related by some function or equation. Simply put, when one variable changes, how does it affect the other?

In our textbook exercise, we're not just looking at how fast the length and width of a rectangle change, but also how these changes impact the area, perimeter, and diagonal. It's important to note that related rates problems often involve the chain rule—a key method in calculus—which allows us to differentiate composite functions.

To solve related rates problems, it's crucial to establish relationships between variables (usually through an equation), differentiate with respect to time, and then substitute known values to find the unknown rate of change. This process emphasizes the interconnectedness of variables and how changes in one aspect lead to changes in others.

Calculus in Geometry

Calculus is a powerful branch of mathematics that extends far beyond simple measurements in geometry into understanding dynamic changes within geometric figures. It can be applied to find tangents to curves, calculate areas under curves, and even in the determination of maximum or minimum points of geometric shapes—a process called optimization.

In our rectangle scenario where length and width are changing over time, calculus, specifically derivative processes, is employed to evaluate the instantaneous rates of change of the area, perimeter, and diagonal. This application is incredibly useful for problems where geometry is not static and dimensions can evolve over time, influenced by various factors.

Understanding calculus in the realm of geometry allows us to tackle a wide range of practical problems, from architectural design where load and stress may alter structures over time to technology advancements where dynamic dimensions play a pivotal role.

Pythagoras' Theorem

Pythagoras' theorem is an ancient, yet eternally practical, geometric principle stating that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem is symbolically represented as \( a^2 + b^2 = c^2 \).

In our exercise, Pythagoras' theorem provides the relationship between the sides of the rectangle (length and width) and the diagonal, with the diagonal acting as the hypotenuse of a right-angled triangle formed by length and width. As length and width change over time, we can use derivatives to assess how those changes affect the diagonal, turning a static geometric equation into a dynamic understanding of spatial relationships.

Beyond its use in static calculations, Pythagoras' theorem integrated with calculus becomes a potent tool in evaluating real-time changes in various fields such as navigation, astronomy, and even in construction, where varying lengths can impact design and functionality.