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Chapter 12: Problem 7

Consider a circle with radius 1 . (a) What percentage of the points lies within \(\frac{1}{2}\) unit of the center? (b) Let \(c\) be a constant with \(0

Short Answer

Expert verified
(a) 25%. (b) \(100c^2\)\%.

Step by step solution

01

Determine the area of the circle

The area of a circle is given by the formula \(\text{Area} = \pi r^2\). Since the radius \(\r\) is 1, the area of the circle is \(\text{Area} = \pi (1)^2 = \pi\).
02

Calculate the area of the smaller circle (part a)

For part (a), we need the area of a smaller circle with radius \(\frac{1}{2}\). The area is \(\text{Area} = \pi (\frac{1}{2})^2 = \pi \frac{1}{4}\).
03

Find the percentage of points within \(\frac{1}{2}\) unit (part a)

To find the percentage, we divide the area of the smaller circle by the area of the larger circle and multiply by 100: \(\text{Percentage} = \frac{\text{Area of smaller circle}}{\text{Area of larger circle}} \times 100 = \frac{\frac{1}{4}\pi}{\pi} \times 100 = \frac{1}{4} \times 100 = 25\%\).
04

Calculate the area for any constant \(c\) (part b)

For part (b), we need the area of a smaller circle with radius \(c\). The area is \(\text{Area} = \pi c^2\).
05

Find the percentage of points within \(c\) unit (part b)

To find the percentage, we divide the area of the smaller circle by the area of the larger circle and multiply by 100: \(\text{Percentage} = \frac{\text{Area of smaller circle}}{\text{Area of larger circle}} \times 100 = \frac{\text{\pi c^2}}{\text{\pi}} \times 100 = c^2 \times 100\%\).

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Key Concepts

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

Circle Area Calculation
The area of a circle is one of the foundational topics in geometry. To calculate the area, we use the formula \( \text{Area} = \pi r^2 \). This formula states that the area is equal to \pi \ times the radius squared. \br\ For example, if you have a circle with a radius of 1 unit, the area is simply \pi (1)^2 = \pi \ unit squares. \br\ If we change the radius to \ \frac{1}{2} \, the area calculation is \pi (\frac{1}{2})^2 = \pi \frac{1}{4}\ unit squares. By understanding this formula, you can find the area for any circle as long as you know its radius.
Percentage Calculation
Percentage calculations are useful for comparing parts of a whole. In this exercise, we need to find what percentage of the points in a larger circle fall within a smaller circle. \br\ To calculate this, you divide the area of the smaller circle by the area of the larger circle, then multiply by 100. \br\ For instance, if the area of a smaller circle is \( \frac{\text{\pi}}{4} \) units and the area of the larger circle is \
circle area calculation
The area of a circle is one of the foundational topics in geometry. To calculate the area, we use the formula \( \text{Area} = \pi r^2 \). This formula states that the area is equal to \pi \ times the radius squared. \br\ For example, if you have a circle with a radius of 1 unit, the area is simply \

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