Problem 65 - Prealgebra

Question

Marta has 60 feet of decorative fencing which she will use for the border of a small circular garden. Find the diameter of the circular garden, correct to the nearest hundredth of a foot. Use ? ? 3.14.

Step-by-Step Solution

Answer

The diameter of the circular garden is approximately 19.11 feet.

Step 1: Understanding the Problem

Marta has a circular garden, and the fence she uses as the border represents the circumference of that circle. We need to find the diameter of the circle. We are given that the circumference is 60 feet.

Step 2: Identify Relevant Formulas

To find the diameter of a circle from its circumference, we use the formula: \( C = \pi \times d \), where \( C \) is the circumference, \( \pi \approx 3.14 \), and \( d \) is the diameter. This can be rearranged to \( d = \frac{C}{\pi} \).

Step 3: Substitute Known Values

Substitute the known values into the formula: \( d = \frac{60}{3.14} \).

Step 4: Calculate the Diameter

Perform the division: \( 60 \div 3.14 \approx 19.10828025 \).

Step 5: Round to the Nearest Hundredth

Round the result, \( 19.10828025 \), to the nearest hundredth, which gives \( 19.11 \).

Key Concepts

Diameter Calculation

To determine the diameter of a circle, especially if you know its circumference, there is a straightforward method. The diameter is simply defined as the longest straight line that can be drawn through the center of a circle, connecting two points on its boundary. Knowing the circumference allows us to reverse-engineer the diameter using a well-known formula.
Typically, the formula relating the circumference (C) to the diameter (D) is: \[ C = \pi \times D \]. By rearranging it, to specifically calculate the diameter, we get: \[ D = \frac{C}{\pi} \].
In other words, if you have the total length around the circle, dividing it by the constant \( \pi \approx 3.14 \) gives you the diameter. Don't forget this approach when you need to solve similar problems!

Mathematical Formulas

Mathematical formulas serve as essential tools in problem-solving. They are like recipes for baking a calculation cake. Knowing which formula to use is half the battle.
In the context of circles, the formula for the circumference \( C \) involves \( \pi \), a special number approximated as 3.14 in many practical exercises. The common circumference formula used here is: \( C = \pi \times D \).
When converting from the circumference to diameter, you rearrange the formula to \( D = \frac{C}{\pi} \). This conversion is crucial because it allows you to explore other circle dimensions without directly measuring them. Understanding how to manipulate these formulas effectively, such as substituting known values and solving for unknowns, empowers you to tackle a range of problems.

Rounding Numbers

Rounding numbers is an important skill, especially when you're dealing with decimals and need a simpler, approximate value. When you have a result like \( 19.10828025 \), it's usually handy to round it to make it easier to read and work with.
Typically, you round numbers to a specified decimal place, like the nearest hundredth, to maintain consistency and reduce complexity. In the example \( 19.10828025 \), rounding to the nearest hundredth requires looking at the thousandth place (third digit after the decimal). If that digit is 5 or more, you round up. If it's less than 5, you round down.
Therefore, \( 19.10828025 \) becomes \( 19.11 \) when rounded to the nearest hundredth. This not only simplifies expressions but makes them more practical for use in real-world situations.