Problem 132 - Calculus Volume 3

Question

The area of a parallelogram with adjacent side lengths that are \(a\) and \(b\), and in which the angle between these two sides is \(\theta,\) is given by the function \(A(a, b, \theta)=b a \sin (\theta) .\) Find the rate of change of the area of the parallelogram with respect to the following: a. Side \(a\) b. Side \(b\) c. Angle \(\theta\)

Step-by-Step Solution

Answer

a. \( b \sin(\theta) \); b. \( a \sin(\theta) \); c. \( ab \cos(\theta) \)

Step 1: Identify the function for the area of the parallelogram

The area of the parallelogram is given by the formula \(A(a, b, \theta) = ab\sin(\theta)\). This function depends on the side lengths \(a\) and \(b\), and the angle \(\theta\) between them.

Step 2: Calculate the rate of change with respect to side a

To find the rate of change of the area with respect to \(a\), take the partial derivative of \(A\) with respect to \(a\): \[\frac{\partial A}{\partial a} = b \sin(\theta)\].

Step 3: Calculate the rate of change with respect to side b

To find the rate of change of the area with respect to \(b\), take the partial derivative of \(A\) with respect to \(b\): \[\frac{\partial A}{\partial b} = a \sin(\theta)\].

Step 4: Calculate the rate of change with respect to angle theta

For the rate of change with respect to \(\theta\), take the partial derivative of \(A\) with respect to \(\theta\), using the derivative of \(\sin(\theta)\) which is \(\cos(\theta)\): \[\frac{\partial A}{\partial \theta} = ab \cos(\theta)\].

Key Concepts

Rate of Change

The rate of change is a fundamental concept in calculus that tells us how a particular quantity changes as another quantity changes. When dealing with multivariable functions, this is often done using partial derivatives.
In this problem, we're asked to find how the area of a parallelogram, given by the function \(A(a, b, \theta) = ab\sin(\theta)\), changes as we change each of the variables: side \(a\), side \(b\), and the angle \(\theta\).

• Partial derivative with respect to \(a\): When we find \(\frac{\partial A}{\partial a} = b\sin(\theta)\), we see the rate at which the area changes if the length of side \(a\) changes while keeping \(b\) and \(\theta\) constant.
• Partial derivative with respect to \(b\): Similarly, \(\frac{\partial A}{\partial b} = a\sin(\theta)\) shows the change in area with changes in side \(b\).
• Partial derivative with respect to \(\theta\): The expression \(\frac{\partial A}{\partial \theta} = ab\cos(\theta)\) demonstrates how the area changes as the angle \(\theta\) changes.

These derivatives allow us to understand individually how each parameter influences the area, assuming the other parameters remain fixed.

Parallelogram Area

Understanding the area of a parallelogram is crucial in many applications, especially in geometry and physics. A parallelogram can be easily characterized by its base and height, or alternatively by the lengths of its adjacent sides and the angle between them.
The formula for calculating the area of a parallelogram given by two sides of lengths \(a\) and \(b\), and the angle \(\theta\) between them, is \(A = ab\sin(\theta)\). This formula comes from the fact that the height of the parallelogram can be expressed as \(b\sin(\theta)\) when \(b\) is considered the base.
Therefore, the function \(A(a, b, \theta)\) ties together these geometric properties, providing an expression to calculate how alterations in side lengths or angle affect the area.

• The term \(\sin(\theta)\) adjusts the area calculation for how much 'tallness' the given angle \(\theta\) contributes, as it reflects the vertical component of side \(b\).
• The product \(ab\sin(\theta)\) effectively combines these properties, showing that the area is directly proportional to the sine of the angle, thus morphing with it.

Grasping this formula aids in visualizing how parallelograms adapt with changes in their structure, helping with diverse practical problems.

Multivariable Calculus

Multivariable calculus extends the concepts of single-variable calculus to functions of multiple variables. This area of mathematics is particularly powerful in working with functions that change in more than one direction and is pivotal in situations like in the given problem where the area of a parallelogram depends on three variables.
In multivariable calculus, partial derivatives are used to capture how a function changes in response to changes in just one of the variables, while keeping others constant. This is essential when seeking to understand the complex behaviors of systems or functions involving several interacting variables.
The function \(A(a, b, \theta) = ab\sin(\theta)\) is a perfect example of such a multivariable function, where:

• The side lengths \(a\) and \(b\) represent two-dimensional variation.
• The angle \(\theta\) introduces a third dimension of variation, making the function dependent on three distinct dimensions.

Through the use of partial derivatives, we dissect how changes in these dimensions individually influence the area. Multivariable calculus thus provides the tools to navigate and solve problems involving such interconnected variables efficiently.