Problem 30 - Beginning and Intermediate Algebra

Question

The radius of a circle is half its diameter. We can express this with the function \(r(d)=\frac{1}{2} d,\) where \(d\) is the diameter of a circle and \(r\) is the radius. The area of a circle in terms of its radius is \(A(r)=\pi r^{2} .\) Find each of the following and explain their meanings. a) \(r(6)\) b) \(A(3)\) c) \(A(r(d))\) d) \(A(r(6))\)

Step-by-Step Solution

Answer

a) \(r(6) = 3\): The radius of the circle is 3 units when the diameter is 6 units. b) \(A(3) = 9\pi\): The area of the circle is \(9\pi\) square units when the radius is 3 units. c) \(A(r(d)) = \pi (\frac{1}{2}d^{2})\): The area of the circle in terms of diameter is \(\pi (\frac{1}{2}d^{2})\) square units. d) \(A(r(6)) = 18\pi\): The area of the circle is \(18\pi\) square units when the diameter is 6 units.

Step 1: Compute r(6)

To find the radius when the diameter is 6, we'll plug in the value 'd=6' into the given function for radius: \(r(d)=\frac{1}{2}d\). r(6) = \(\frac{1}{2}(6)\) = 3 The radius of the circle is 3 units when the diameter is 6 units. b) Finding the area of a circle with radius 3

Step 2: Compute A(3)

To find the area of the circle when the radius is 3, we'll plug in the value 'r=3' into the given area function: \(A(r)=\pi r^{2}\). A(3) = \(\pi (3)^{2}\) = 9π The area of the circle is \(9\pi\) square units when the radius is 3 units. c) Finding the area of a circle in terms of diameter

Step 3: Compute A(r(d))

To find the area of the circle in terms of diameter, we need to substitute the radius function, \(r(d)=\frac{1}{2}d\), into the area function, \(A(r)=\pi r^{2}\). A(r(d)) = \(\pi [\frac{1}{2}d]^{2}\) = \(\pi (\frac{1}{2}d^{2})\) The area of the circle in terms of diameter is \(\pi (\frac{1}{2}d^{2})\) square units. d) Finding the area of a circle with diameter 6

Step 4: Compute A(r(6))

To find the area of the circle when the diameter is 6, we will use the result from part c and plug in the value 'd=6': A(r(6)) = \(\pi (\frac{1}{2}(6)^{2})\) = \(\pi (\frac{1}{2}(36))\) = 18π The area of the circle is \(18\pi\) square units when the diameter is 6 units.

Key Concepts

Radius

The radius of a circle is a fundamental concept in geometry. It is defined as the distance from the center of the circle to any point on its circumference.

The radius is essential for calculating other properties of a circle, such as the diameter and the area. One important relationship to remember is that the radius is always half of the diameter. This means if you know the diameter of a circle, you simply divide it by two to get the radius.

In mathematical terms, if the diameter is represented by 'd', then the radius 'r' can be calculated using the formula:

• \( r = \frac{1}{2}d \)

This formula helps simplify many circle-related problems, especially when calculating the area or circumference of a circle.

Diameter

The diameter of a circle is another core measurement in circle geometry. Imagine drawing a straight line that passes through the center of a circle and touches two points on its edge; this line segment is the diameter.

The diameter is crucial because it represents the largest distance across the circle. Furthermore, it is exactly twice the length of the radius. Therefore, knowing either the radius or the diameter allows you to determine the other. This relationship is expressed mathematically as:

• \( d = 2r \)

Understanding the diameter helps in solving many geometric problems, including calculating the circumference, since the circumference is directly related to the diameter by the formula \( C = \pi d \).

Area of a Circle

The area of a circle is a measure of the space enclosed within its circumference. It is typically measured in square units and can be calculated with a special formula once you know the radius.

The formula to compute the area 'A' of a circle given its radius 'r' is:

• \( A = \pi r^2 \)

This formula shows that the area increases with the square of the radius. For example, doubling the radius results in an area four times larger.

Alternatively, if you know the diameter instead of the radius, you can compute the radius first using \( r = \frac{1}{2}d \) and then calculate the area. This makes it easy to calculate the area no matter which measurement you start with. Understanding this concept is crucial for various applications both in mathematics and real-world scenarios, such as designing circular objects or calculating land areas.