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Chapter 1: Problem 26

Express the area of a circle as a function of: (a) its diameter, \(d\). (b) its circumference, \(c .\)

Short Answer

Expert verified
The area of a circle can be expressed as a function of its diameter as \( A = \frac{\pi d^2}{4} \) and as a function of its circumference as \( A = \frac{c^2}{4\pi} \).

Step by step solution

01

Express Area in terms of Diameter

Start by using the relation between diameter and radius \( d = 2r \). Solve this equation for \( r \), which gives us \( r = d/2 \). Substitute this into the circle area formula \( A = \pi r^2 \) to get \( A = \pi (d/2)^2 \), which simplifies to \( A = \frac{\pi d^2}{4} \).
02

Express Area in terms of Circumference

Next, use the relation between circumference and radius \( c = 2\pi r \). Solve this equation for \( r \), giving \( r = c / (2\pi) \). Substitute this into the circle area formula \( A = \pi r^2 \) to get \( A = \pi [c / (2\pi)]^2 \), which simplifies to \( A = \frac{c^2}{4\pi} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Circle Area Formula
Understanding how to find the area of a circle is fundamental in geometry and is frequently applied in various real-life scenarios. The circle area formula is given by \( A = \pi r^2 \), where \( A \) represents the area and \( r \) is the radius. The radius is the distance from the center of the circle to its boundary. Since \pi (pi) is a constant approximately equal to 3.14159, the key to finding the area lies in knowing the radius of the circle.

To make it even simpler, imagine a circle as a pie. The area formula helps you calculate how much space that pie takes up on your plate, based on the length from the middle of the pie to its edge鈥攖he radius. If you know the radius, you can easily compute how much space your circular pie or any circle occupies by just plugging that number into the formula.
Diameter and Radius Relation
When working with circles, it's essential to understand the relationship between the diameter and the radius, as they are both key measurements. The diameter of a circle is the straight line passing through the center and touching two points on its edge. It's symbolized by \( d \).

The radius鈥攕ymbolized by \( r \)鈥攊s half the length of the diameter, which leads to the straightforward relationship: \( d = 2r \). Conversely, if you have the diameter, you can find the radius by halving it, expressed as \( r = \frac{d}{2} \).

For instance, if a circle's diameter is 10 units, then its radius would be 5 units. This relationship is critical because it allows us to move between two different properties of a circle easily. When you have the diameter but need to calculate the area, you can convert diameter to radius with this relation and then apply the circle area formula.
Circumference and Radius Relation
Another essential concept in circle geometry is the relationship between the circumference and the radius. The circumference is the perimeter or the distance around the circle. It is denoted by \( c \). The formula connecting the circumference with the radius is \( c = 2\pi r \).

This means that if you know the circumference, you can determine the radius by rearranging the formula to \( r = \frac{c}{2\pi} \). This relationship is vital when you're given the circumference and need to find the circle鈥檚 area. After finding the radius with this formula, you can plug it into the circle area formula to calculate the area.

For example, if a wheel's circumference is 31.4 units, the radius of the wheel would be 5 units because 31.4 divided by \(2\pi\) (approximately 6.28) equals 5. Knowing the circumference and radius relation helps expand our toolkit for solving various circle-related problems.

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Most popular questions from this chapter

For each function, determine the largest possible domain. (a) \(f(x)=\frac{3}{x}+\frac{2}{3-x}-\frac{x}{2 x+2}\) (b) \(g(x)=\sqrt{x}+2 \sqrt{3-x}\)

Draw the graph of a function \(f\) that is 1 -to- 1 and a function \(g\) that is not 1 -to-1.

(a) \(f(x)=\frac{1}{x+2}\) (b) \(g(x)=\frac{5}{x^{2}+4}\)

A typist can type \(W\) words per minute. On average, each computer illustration takes \(C\) minutes to create and \(I\) minutes to insert. (a) What is the estimated amount of time it will take for this typist to create a document \(N\) words long and containing \(Z\) illustrations? (b) The typist is paid \(\$ 13\) per hour for typing and a flat rate of \(\$ 10\) per picture. The cost of getting a document typed is a function of its length and the number of pictures. Write a function that gives a good estimate of the cost of getting a document of \(x\) words typed, assuming that the ratio of illustrations to words is \(1: 1000\). (c) Given the document described in part (b), express the typist's wages per hour as a function of \(x\).

Two sisters, Nina and Lori, part on a street corner. Lori saunters due north at a rate of 150 feet per minute and Nina jogs off due east at a rate of 320 feet per minute. Assuming they maintain their speeds and directions, express the distance between the sisters as a function of the number of minutes since they parted.
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