Problem 18 - University Calculus: Early Transcendentals

Question

If \(x, y,\) and \(z\) are lengths of the edges of a rectangular box, the common length of the box's diagonals is \(s=\) \(\sqrt{x^{2}+y^{2}+z^{2}}\). a. Assuming that \(x, y,\) and \(z\) are differentiable functions of \(t\) how is \(d s / d t\) related to \(d x / d t, d y / d t,\) and \(d z / d t ?\) b. How is \(d s / d t\) related to \(d y / d t\) and \(d z / d t\) if \(x\) is constant? c. How are \(d x / d t, d y / d t,\) and \(d z / d t\) related if \(s\) is constant?

Step-by-Step Solution

Answer

a. \( \frac{ds}{dt} = \frac{x \frac{dx}{dt} + y \frac{dy}{dt} + z \frac{dz}{dt}}{\sqrt{x^2 + y^2 + z^2}} \). b. \( \frac{ds}{dt} = \frac{y \frac{dy}{dt} + z \frac{dz}{dt}}{\sqrt{x^2 + y^2 + z^2}} \) if \( x \) is constant. c. \( x \frac{dx}{dt} + y \frac{dy}{dt} + z \frac{dz}{dt} = 0 \) if \( s \) is constant.

Step 1: Differentiate with respect to time

Given the equation for the diagonal of the box, which is \( s = \sqrt{x^2 + y^2 + z^2} \), differentiate both sides with respect to time \( t \). Use the chain rule to obtain an expression for \( \frac{ds}{dt} \): \[ \frac{d}{dt}(s) = \frac{d}{dt}\left(\sqrt{x^2 + y^2 + z^2}\right). \] Applying the chain rule gives: \[ \frac{ds}{dt} = \frac{1}{2\sqrt{x^2 + y^2 + z^2}} \cdot 2\left(x\frac{dx}{dt} + y\frac{dy}{dt} + z\frac{dz}{dt}\right). \] Simplify this to: \[ \frac{ds}{dt} = \frac{x \frac{dx}{dt} + y \frac{dy}{dt} + z \frac{dz}{dt}}{\sqrt{x^2 + y^2 + z^2}}. \]

Step 2: Special case when x is constant

If \( x \) is constant, \( \frac{dx}{dt} = 0 \). Substituting this into the differentiated equation gives: \( \frac{ds}{dt} = \frac{x \cdot 0 + y \frac{dy}{dt} + z \frac{dz}{dt}}{\sqrt{x^2 + y^2 + z^2}}. \) This simplifies to: \( \frac{ds}{dt} = \frac{y \frac{dy}{dt} + z \frac{dz}{dt}}{\sqrt{x^2 + y^2 + z^2}}. \)

Step 3: Condition for constant diagonal length

If the length of the diagonal \( s \) is constant over time, then \( \frac{ds}{dt} = 0 \). From the expression obtained in Step 1, this implies: \[ x \frac{dx}{dt} + y \frac{dy}{dt} + z \frac{dz}{dt} = 0. \] This equation represents a relationship between the rates of change of \( x, y, \) and \( z \) when the diagonal is of constant length.

Key Concepts

Differentiation

Differentiation is a powerful tool in calculus used to determine how a function changes as its input changes. It involves finding the derivative, which represents the function's rate of change. In the context of our exercise, differentiation helps us understand how the diagonal length of a rectangular box changes over time when the box's dimensions are functions of time.
For example, if we want to know how fast the diagonal of the box changes as its sides change, we differentiate the expression for the diagonal with respect to time.Here's a quick breakdown of the process:

• Identify the function: For the diagonal, the function is given by the equation \( s = \sqrt{x^2 + y^2 + z^2} \).
• Differentiation with respect to time \( t \): We apply the derivative operator \( \frac{d}{dt} \) to this function.
• The result is a new equation that shows the relationship between \( \frac{ds}{dt} \) and the rates of change \( \frac{dx}{dt}, \frac{dy}{dt}, \) and \( \frac{dz}{dt} \).

Chain Rule

The chain rule is a fundamental theorem in calculus that allows us to differentiate composite functions. It's especially useful when dealing with functions where variables depend on other variables, like in our problem where \( x, y, \) and \( z \) are functions of time.When we differentiate the diagonal equation \( s = \sqrt{x^2 + y^2 + z^2} \), we apply the chain rule to deal with the square root function. The chain rule states that:

• You first differentiate the outer function (in this case, the square root), and then multiply by the derivative of the inner function (the expression under the square root).

This would look like:\[\frac{ds}{dt} = \frac{1}{2\sqrt{x^2 + y^2 + z^2}} \times 2\left( x \frac{dx}{dt} + y \frac{dy}{dt} + z \frac{dz}{dt} \right). \]The factor of \( 2 \) cancels out, which simplifies the result to:\[\frac{ds}{dt} = \frac{x \frac{dx}{dt} + y \frac{dy}{dt} + z \frac{dz}{dt}}{\sqrt{x^2 + y^2 + z^2}}.\]This equation clearly shows how each individual rate of change impacts the rate of change of the diagonal.

Rectangular Box

A rectangular box, also known as a cuboid, is a three-dimensional shape with six rectangular faces. The edges of the box, represented by \( x, y, \) and \( z \), form the sides of these faces. Understanding the dimensions of the box—and how they change—helps in determining various geometric properties, including the diagonal's length.In this exercise, our focus is on how the dimensions of the box change relative to each other and over time:

• Each variable \( x, y, \) and \( z \) is considered a differentiable function of time \( t \).
• These dimensions interact, affecting how properties like the diagonal length \( s \) change over time.

When one side remains constant, as explored in the exercise's part (b), we can focus on how changes in the other two dimensions affect the diagonal. For instance, if \( x \) is constant, then changes in \( y \) and \( z \) determine how \( s \) varies.

Diagonals

The diagonal of a rectangular box is a segment that connects two opposite corners through the box's interior. Its length \( s \) can be found using the Pythagorean theorem in three dimensions: \[s = \sqrt{x^2 + y^2 + z^2}\]This diagonal length is crucial because it links the box's spatial dimensions directly.In related rates problems, the diagonal often acts as a composite function involving the box's length, width, and height:

• The diagonal offers insight into how the overall size of the box changes even if each dimension changes differently.
• In the exercise, related rates of \( x, y, \) and \( z \) help us understand how \( s \) itself is changing with time, as in \( \frac{ds}{dt} \).
• Scenarios like constant \( x \) or constant diagonal \( s \) show how specific conditions lead to particular relationships between the rates of change of each dimension.

Through such analysis, the diagonal becomes a window into the dynamic aspects of a box's geometry.