Problem 25
Question
Area of Circle What is the rate of change of the area of a circle with respect to the radius when the radius is \(r=3\) in. ?
Step-by-Step Solution
Verified Answer
The rate of change of the area of the circle with respect to the radius when the radius is 3 inches is \(6\pi\) square inches per inch.
1Step 1: Write down the formula for the area of a circle
The Area \(A\) of a circle with radius \(r\) is given by the formula \(A = \pi r^2\).
2Step 2: Differentiate the area formula with respect to radius
Differentiating both sides of the equation with respect to \(r\) gives us the rate of change of the area with respect to the radius, which is \( \frac{dA}{dr} = 2\pi r \)
3Step 3: Substitute the given radius into the derivative
Now, substitute the given value \(r = 3\) into the derivative \( \frac{dA}{dr} = 2\pi r \). Hence, when \(r = 3\), \( \frac{dA}{dr} = 2\pi \times 3 = 6\pi \) square inches per inch.
Key Concepts
DifferentiationArea of Circle FormulaApplying Derivatives to Geometry
Differentiation
Differentiation is a fundamental concept in calculus that deals with how a function changes as its input changes. It is the process of finding the derivative of a function, which provides the rate at which the function value changes concerning its variable.
For example, if you have a function that represents the position of a moving object over time, differentiation allows you to find the object's velocity by calculating the derivative of the position with respect to time. This gives you the rate of change of the position, essentially telling you how fast the object is moving and whether its speed is increasing or decreasing.
The differentiation process uses specific rules to calculate derivatives efficiently. These rules include the power rule, which states that the derivative of a function of the form \( x^n \) is \( nx^{n-1} \) where \( n \) is a real number. This rule is crucial when finding the rate of change in geometrical shapes, such as the area of a circle with respect to its radius.
For example, if you have a function that represents the position of a moving object over time, differentiation allows you to find the object's velocity by calculating the derivative of the position with respect to time. This gives you the rate of change of the position, essentially telling you how fast the object is moving and whether its speed is increasing or decreasing.
The differentiation process uses specific rules to calculate derivatives efficiently. These rules include the power rule, which states that the derivative of a function of the form \( x^n \) is \( nx^{n-1} \) where \( n \) is a real number. This rule is crucial when finding the rate of change in geometrical shapes, such as the area of a circle with respect to its radius.
Area of Circle Formula
The next core concept is the area of a circle formula, which is a critical piece of knowledge for any geometry and calculus student. The formula to calculate the area of a circle is \( A = \pi r^2 \), where \( A \) represents the area, and \( r \) is the radius of the circle.
This equation succinctly tells us that the area of a circle is proportional to the square of its radius. As the radius grows or shrinks, the area changes, but not in a linear fashion; instead, it changes according to a square law. This observation is critical since it affects how we perceive rates of change in geometric figures. Understanding how to apply this formula and manipulate the variables within it is essential for solving a variety of problems, including predicting how changes in the radius affect the overall size of the circle.
This equation succinctly tells us that the area of a circle is proportional to the square of its radius. As the radius grows or shrinks, the area changes, but not in a linear fashion; instead, it changes according to a square law. This observation is critical since it affects how we perceive rates of change in geometric figures. Understanding how to apply this formula and manipulate the variables within it is essential for solving a variety of problems, including predicting how changes in the radius affect the overall size of the circle.
Applying Derivatives to Geometry
Finally, applying derivatives to geometry allows us to understand and calculate the dynamic relationships and rates of change within geometrical shapes. By combining the concept of differentiation with geometrical formulas, like that of a circle's area, you can determine how these measures change in response to one another.
For instance, when we apply differentiation to the area of a circle formula \( A = \pi r^2 \), we obtain \( \frac{dA}{dr} = 2\pi r \), which expresses how the area of the circle changes with respect to its radius. This derivative is known as 'the rate of change of area concerning the radius.' In practical terms, this allows us to calculate how much the area will increase or decrease given a change in radius, which is a common problem in various fields such as engineering, physics, and even economics where one might consider spaces or regions related to circular shapes.
Understanding this application of derivatives to geometry not only enhances problem-solving skills but also enriches comprehension of the mathematical relationships inherent in the physical world.
For instance, when we apply differentiation to the area of a circle formula \( A = \pi r^2 \), we obtain \( \frac{dA}{dr} = 2\pi r \), which expresses how the area of the circle changes with respect to its radius. This derivative is known as 'the rate of change of area concerning the radius.' In practical terms, this allows us to calculate how much the area will increase or decrease given a change in radius, which is a common problem in various fields such as engineering, physics, and even economics where one might consider spaces or regions related to circular shapes.
Understanding this application of derivatives to geometry not only enhances problem-solving skills but also enriches comprehension of the mathematical relationships inherent in the physical world.
Other exercises in this chapter
Problem 24
In Exercises \(19 - 28 ,\) determine the limit graphically. Confirm algebraically. $$\lim _ { x \rightarrow 0 } \frac { \sin 2 x } { x }$$
View solution Problem 24
In Exercises \(21-26,\) find \(\lim _{x \rightarrow \infty} y\) and \(\lim _{x \rightarrow-\infty} y\). $$y=\frac{2 x+\sin x}{x}$$
View solution Problem 25
In Exercises \(19 - 28 ,\) determine the limit graphically. Confirm algebraically. $$\lim _ { x \rightarrow 0 } \frac { \sin x } { 2 x ^ { 2 } - x }$$
View solution Problem 25
In Exercises \(21-26,\) find \(\lim _{x \rightarrow \infty} y\) and \(\lim _{x \rightarrow-\infty} y\). $$y=\frac{\sin x}{2 x^{2}+x}$$
View solution