Question

City officials rope off a circular area to prepare for a concert in the park. They estimate that each person occupies 6 square feet. Describe how you can use a radical inequality to determine the possible radius of the region when \(P\) people are expected to attend the concert.

Short answer

To fit \(P\) people in the circular region, the radius \(r\) of the circle must be at least \(\sqrt{\frac{6P}{\pi}}\) feet, given that each person requires 6 square feet of space. This can be represented by the inequality \(r \geq \sqrt{\frac{6P}{\pi}}\).

Step-by-step solution

Establish the Relationship

First, establish a relationship between the area of the circle and the space required for each person. Use the formula for the area of a circle, which is \(\pi r^2\). Since the area should be equal to or larger than the space required for each attendee, we can write the inequality as \(\pi r^2 \geq 6P\), where \(P\) is the number of attendees.

Solve for r

Next, we want to solve for \(r\). Divide both sides of the inequality by \(\pi\) to isolate \(r^2\), which yields: \(r^2 \geq \frac{6P}{\pi}\). To solve for \(r\), take the square root of both sides. Remember that the square root of a number is always positive, so we get: \(r \geq \sqrt{\frac{6P}{\pi}}\). Remember, we used the square root because \(r^2\) was isolated on the one side of the inequality.

Interpret the Result

The result \(r \geq \sqrt{\frac{6P}{\pi}}\) gives the radius of the circle. This inequality represents the fact that the radius of the circle must be at least \(\sqrt{\frac{6P}{\pi}}\) feet for \(P\) people to attend the concert, given that each person requires 6 square feet of space.

Key concepts

Area of a Circle

The concept of the "area of a circle" is fundamental in geometry and mathematics. The area of a circle is the space contained within its circumference. Think of it as how many square units can fit inside the circle. It's typically measured in square units such as square feet, square meters, etc.

The formula to calculate it is:

• Area = \( \pi r^2 \)

Here, \( \pi \) (pi) is a constant approximately equal to 3.14159, and \( r \) is the radius of the circle.

To find out the area, you simply square the radius (multiply it by itself), then multiply that result by \( \pi \). For instance, if you have a circle with a radius of 3 feet, the area becomes \( 3.14159 \times 3 \times 3 \) which equals roughly 28.27 square feet.

Inequality

In mathematics, an "inequality" is a statement that describes the relationship between one value being less than or greater than another value. Inequalities are often expressed using symbols such as:

• \( > \) which means greater than
• \( < \) which means less than
• \( \geq \) which means greater than or equal to
• \( \leq \) which means less than or equal to

In the context of the concert problem, an inequality helps us understand that the area of the circle must not be smaller than the total space required for all attendees.

The inequality \( \pi r^2 \geq 6P \) reflects that the circle's area should be big enough to accommodate all people, where each person needs 6 square feet. It implies that the calculated area (left side of the inequality) could be more than or equal to the required area (right side of the inequality) but never less.

Radius

The "radius" of a circle is the distance from the center of the circle to any point on its boundary (circumference). It's the essential measure needed to grasp other circle-related concepts like diameter or area.

The radius is half of the diameter of the circle, and it plays a crucial role in calculating the circle's area. You can visualize the radius as the spoke on a bicycle wheel, extending from the hub (center) to the rim (edge).

To determine the radius for a space requirement in the concert problem, you find the smallest acceptable radius using the inequality solution: \( r \geq \sqrt{\frac{6P}{\pi}} \). This ensures the circle is large enough for all attendees to fit comfortably.

Square Root

The "square root" is a mathematical function that finds the original number which, when multiplied by itself, gives you a particular value. It is symbolized by the radical sign \( \sqrt{} \).

• If \( x^2 = a \), then \( x = \sqrt{a} \).

The square root helps in reversing the squaring process. For example, since \( 4^2 = 16 \), \( \sqrt{16} = 4 \).

In the concert scenario, taking the square root of \( \frac{6P}{\pi} \) allows us to solve for the radius \( r \) in the formula \( r^2 \). By finding the square root, you determine the minimum radius needed to ensure everyone has enough space. It is key to unraveling the inequality because it converts \( r^2 \geq \) into \( r \geq \sqrt{\frac{6P}{\pi}} \), providing a clear condition for the circle's size.