Problem 25 - A Graphical Approach to College Algebra

Question

Solve each problem. Radius Covered by a Circular Lawn Sprinkler A square lawn has area 800 square feet. A sprinkler placed at the center of the lawn sprays water in a circular pattern that just covers the lawn. What is the radius of the circle?

Step-by-Step Solution

Answer

The radius of the circle is approximately 14.14 feet.

Step 1: Understand the Problem

A circular sprinkler is placed at the center of a square lawn with an area of 800 square feet. The circle created by the sprinkler covers exactly the entire lawn.

Step 2: Find the Side Length of the Square

The area of a square is calculated using the formula: \( \text{Area} = s^2 \) where \( s \) is the side length. Given the area is 800 square feet, solve for \( s \): \[ s^2 = 800 \] \[ s = \sqrt{800} \] \[ s = \sqrt{100 \times 8} \] \[ s = 10\sqrt{8} \] \[ s \approx 28.28 \text{ feet} \]

Step 3: Determine the Radius of the Circle

The circle from the sprinkler covers the entire square, meaning its diameter is equal to the side of the square. Therefore, the radius \( r \) of the circle, which is half of the diameter, is calculated as:\[ r = \frac{s}{2} \] \[ r = \frac{28.28}{2} \] \[ r \approx 14.14 \text{ feet} \]

Key Concepts

Area of square

The area of a square is a straightforward concept in geometry. It determines the amount of space within the boundaries of a square. The formula for the area is given as:

• \( \text{Area} = s^2 \)
• where \( s \) is the length of one side of the square.

To visualize this, imagine a square divided into small equal-sized squares within, each 1 unit by 1 unit. The total count of these squares will give you the area.
For example, if one side of a square lawn is \( s = 4 \) feet, then its area would be \( s^2 = 16 \) square feet. Knowing the area and calculating the side length involves working with square roots, which is discussed next.

Square roots

Square roots are fundamental in solving various mathematical problems, especially those involving squares. The square root of a number \( x \) is a value that, when multiplied by itself, gives \( x \).

• For any positive number \( x \), \( \sqrt{x} \times \sqrt{x} = x \).

To solve for the side of a square when you have its area, you use its square root. For instance, with an area of 800 square feet, to find the side \( s \) of the square:

• \( s^2 = 800 \) implies \( s = \sqrt{800} \).

Simplifying \( \sqrt{800} \) involves breaking it down: \( 800 = 100 \times 8 \), so \( \sqrt{800} = \sqrt{100} \times \sqrt{8} \). This results in \( 10\sqrt{8} \), and approximately \( s \approx 28.28 \) feet.

Radius of circle

Understanding the radius of a circle is a vital part of solving problems involving circular shapes. The radius is simply the distance from the center of the circle to any point on its edge.

• A circle's diameter is twice its radius.
• Thus, \( \text{Radius} = \frac{\text{Diameter}}{2} \).

When a sprinkler sprays water in a circular pattern that exactly covers a square lawn, the circle's diameter must equal a side of the square. This means if the square's side length is 28.28 feet, the radius \( r \) is \( \frac{28.28}{2} \variant{check_this}\), approximately 14.14 feet. This radius ensures that the sprinkler covers the entire lawn surface efficiently.

Diameter

The diameter of a circle is an important measurement as it defines the circle's width from one side to the other through the center. It is twice the length of the radius.

• \( \text{Diameter} = 2 \times \text{Radius} \)
• Alternatively, \( \text{Diameter} = \text{side length} \) for circles embedded in squares.

Considering our sprinkler problem, the circle's diameter matches the side length of the square lawn, which is 28.28 feet. Knowing this, you can easily compute the circle's radius by halving the diameter, which helps determine how far the sprinkler can effectively cover the lawn.