Problem 160 - Calculus 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 in/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 is changing at a rate of approximately 3.605 in./sec.

Step 1: Understanding the Problem

We need to find how fast the diagonal of a rectangle is changing when the sides are changing. The lengths of the sides are given, and we know the rates at which these side lengths are changing.

Step 2: Apply Pythagorean Theorem

Use the Pythagorean theorem to express the diagonal of the rectangle in terms of its sides. If \( a \) and \( b \) represent the lengths of the sides, and \( d \) is the diagonal, then \( d = \sqrt{a^2 + b^2} \).

Step 3: Differentiate with Respect to Time

Differentiate the equation \( d = \sqrt{a^2 + b^2} \) with respect to time \( t \). Applying the chain rule, we get \( \frac{dd}{dt} = \frac{1}{2\sqrt{a^2 + b^2}} \cdot 2a \frac{da}{dt} + 2b \frac{db}{dt} \). This simplifies to \( \frac{dd}{dt} = \frac{a \frac{da}{dt} + b \frac{db}{dt}}{\sqrt{a^2 + b^2}} \).

Step 4: Plug in Known Values

Substitute \( a = 16 \) in., \( b = 20 \) in., \( \frac{da}{dt} = 2 \) in./sec, and \( \frac{db}{dt} = 4 \) in./sec into the differentiated equation. Calculate the diagonal \( d = \sqrt{16^2 + 20^2} \).

Step 5: Calculate the Rate of Change of the Diagonal

Using the formula \( \frac{dd}{dt} = \frac{16 \times 2 + 20 \times 4}{\sqrt{16^2 + 20^2}} \), compute the value to find how fast the diagonal is changing with respect to time.

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental concept in geometry. It relates the lengths of the sides of a right triangle to its hypotenuse. The theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Mathematically, if we have a triangle with sides of length \(a\) and \(b\), and a hypotenuse of length \(c\), then:\[c^2 = a^2 + b^2\]In the context of this exercise, we use the sides of a rectangle. Each pair of adjacent sides forms a right triangle with the diagonal being the hypotenuse. By substituting the actual side measurements, we find the length of the diagonal using:\[d = \sqrt{a^2 + b^2}\]
In our exercise, with values \(a = 16\) and \(b = 20\), we calculate:\[d = \sqrt{16^2 + 20^2} = \sqrt{256 + 400} = \sqrt{656}\]Thus, the length of the diagonal when the sides are \(16\) inches and \(20\) inches is approximately \(25.61\) inches.

Differentiation

Differentiation is a key concept in calculus that deals with the rate of change of quantities. When we have a changing value, such as the side of a rectangle growing over time, differentiation helps us understand how another related quantity changes. In this problem, we differentiate the equation representing the diagonal of a rectangle to find how quickly the diagonal itself changes when the sides vary over time. The original equation derived from the Pythagorean Theorem is:\[d = \sqrt{a^2 + b^2}\]By differentiating both sides with respect to time \(t\), we find the rate at which the diagonal changes over time. This provides us with a relationship between how fast each side of the rectangle changes and how fast the diagonal changes.

Chain Rule

The Chain Rule in calculus is an essential method used to differentiate composite functions – functions made up of multiple layers. It allows us to find the derivative of such functions by breaking them down into simpler components. When we have the equation \(d = \sqrt{a^2 + b^2}\), we consider \(a\) and \(b\) as functions of time. Hence, the diagonal \(d\) indirectly depends on time. To differentiate \(d\) with respect to time, we apply the chain rule:\[\frac{dd}{dt} = \frac{1}{2\sqrt{a^2 + b^2}} \times (2a \frac{da}{dt} + 2b \frac{db}{dt})\]This can be simplified to:\[\frac{dd}{dt} = \frac{a \cdot \frac{da}{dt} + b \cdot \frac{db}{dt}}{\sqrt{a^2 + b^2}}\]
In this context, each term within the formula is crucial as it represents how changes in the individual sides affect the diagonal's rate of change. Once we substitute the known values of \(a\), \(b\), \(\frac{da}{dt}\), and \(\frac{db}{dt}\), we can solve for \(\frac{dd}{dt}\), providing us with the speed at which the diagonal is changing.