Problem 327 - Calculus Volume 1

Question

For the following exercises, consider a closed rectangular box with a square base with side \(x\) and height \(y\) If the surface area of the rectangular box is 78 square feet, find \(\frac{d y}{d x}\) when \(x=3\) feet and \(y=5\) feet.

Step-by-Step Solution

Answer

\(\frac{dy}{dx} = \frac{-13}{6}\) when \(x = 3\) and \(y = 5\).

Step 1: Write the Surface Area Formula

A rectangular box with a square base has a surface area given by: \[S = x^2 + 4xy\]where \(x^2\) is the area of the base and \(4xy\) is the area of the four vertical sides.

Step 2: Set Up the Given Condition

According to the exercise, the surface area \(S\) is 78 square feet:\[x^2 + 4xy = 78\]This equation will be used to find expressions for the derivatives.

Step 3: Implicit Differentiate with Respect to x

Differentiate both sides of the equation \(x^2 + 4xy = 78\) implicitly with respect to \(x\):\[2x + 4x \frac{dy}{dx} + 4y = 0\]This needs to be solved for \(\frac{dy}{dx}\).

Step 4: Solve for dy/dx

Rearrange the differentiated equation to solve for \(\frac{dy}{dx}\):\[4x \frac{dy}{dx} = -2x - 4y\]Divide both sides by \(4x\) to isolate \(\frac{dy}{dx}\):\[\frac{dy}{dx} = \frac{-2x - 4y}{4x} = \frac{- (2x + 4y)}{4x}\]

Step 5: Substitute Given Values

Substitute \(x = 3\) and \(y = 5\) into the expression for \(\frac{dy}{dx}\):\[\frac{dy}{dx} = \frac{- (2(3) + 4(5))}{4(3)} = \frac{- (6 + 20)}{12} = \frac{-26}{12}\]Simplify the fraction:\[\frac{dy}{dx} = \frac{-13}{6}\]

Key Concepts

Implicit Differentiation

In calculus, implicit differentiation is a powerful technique used for finding the derivatives of equations that are not solved for one specific variable. When an equation involves multiple variables, differentiating explicitly with respect to one variable may not always be straightforward. Implicit differentiation helps us find the derivative when one variable depends on another in a more indirect way.

Instead of rearranging the equation to solve for a particular variable, which might not always be feasible or simple, implicit differentiation allows us to differentiate both sides of the equation directly with respect to a particular variable. The key benefit here is maintaining the relationship between the variables throughout the differentiation process.

For example, when differentiating the equation for a rectangular box's surface area, given by \(x^2 + 4xy = 78\), with respect to \(x\), we apply the derivative on each term individually. The term \(x^2\) differentiates to \(2x\), while \(4xy\) requires the product rule, resulting in \(4x \frac{dy}{dx} + 4y\). By treating \(y\) as a function of \(x\), and thus introducing \(\frac{dy}{dx}\) as part of our differentiated expression, we leverage implicit differentiation to solve for \(\frac{dy}{dx}\) without explicitly solving for \(y\).

• Efficient for multi-variable relationships.
• Simplifies differentiation of complex equations.
• Directly explores variable interdependencies.

Surface Area Calculations

The surface area of a three-dimensional object is a measure of the total area that the surface of the object occupies. In practical applications, calculating surface area is crucial in fields like architecture, engineering, and manufacturing.

For a closed rectangular box with a square base, the surface area can be calculated by finding the sum of the areas of all its sides. Specifically for such a box, the surface area \(S\) is calculated as \(x^2 + 4xy\), where:

• \(x^2\) is the area of the square base.
• \(4xy\) represents the collective area of the four vertical rectangular sides.

Each of the four sides of the box is a rectangle with the base as one side of length \(x\) and height \(y\). Multiplying these dimensions gives the individual area of one side, which is then multiplied by four, as there are four identical sides in such a setup.

Understanding these calculations helps in designing structures and packaging, ensuring efficient material usage to cover and protect objects completely while minimizing waste. This primary knowledge is also beneficial for solving complex calculus problems involving such geometric shapes, as seen in the given exercise.

Rectangular Box Geometry

Geometry of rectangular boxes not only encompasses length, width, and height but also extends to concepts like volume and surface area. When the base of the box is square, this simplifies some calculations but also introduces unique relationships between the dimensions.

A rectangular box with a square base has its own features:

• The shortest two dimensions are equal: \(x = \text{side of the square base}\).
• The third dimension is the height \(y\), which can vary independently.

This setup provides symmetrical properties and simplifies calculations for properties like volume and surface area. The volume \(V\) is given by \(x^2y\), while the surface area \(S\) is calculated by \(x^2 + 4xy\) in the context of this problem.

Understanding these dimensions and their interactions is fundamental in determining functions like surface areas or derivatives, as well as applying implicit differentiation. It also allows insight into optimizing box dimensions for various applications, whether it be minimizing material use or maximizing storage space. This type of problem-solving is essential in both theoretical studies and real-life pragmatic problems.