The circumference of a circular garden is 100.48 feet. What is the diameter of the garden? Use 3.14 for π and do not round your answer.
The Correct Answer and Explanation is:
To find the diameter of a circular garden when the circumference is given, we can use the formula for the circumference of a circle:
[
C = \pi \times d
]
where ( C ) is the circumference and ( d ) is the diameter of the circle. Given that the circumference ( C ) is 100.48 feet and using ( \pi \approx 3.14 ), we can rearrange the formula to solve for the diameter:
[
d = \frac{C}{\pi}
]
Substituting the given values into the formula, we have:
[
d = \frac{100.48}{3.14}
]
Calculating this gives:
[
d = 32.0 \text{ feet}
]
Therefore, the diameter of the garden is 32.0 feet.
Explanation
The relationship between the circumference and diameter of a circle is fundamental in geometry. The circumference is the distance around the circle, while the diameter is the longest distance across the circle, passing through the center. The formula ( C = \pi \times d ) illustrates how these two quantities are related, with ( \pi ) serving as a constant that expresses the ratio of the circumference to the diameter.
In our example, we started with a known circumference of 100.48 feet. By using the approximation of ( \pi ) as 3.14, we could easily isolate the diameter in our calculations. This approach is often used in real-world applications, such as landscaping, engineering, and architecture, where circular measurements are common.
Understanding how to manipulate these formulas allows for effective design and accurate measurements in circular constructions. The calculation illustrates the practical application of mathematical concepts in everyday life, especially in fields requiring precision, such as horticulture when determining the space needed for planting, or in construction projects where circular features are involved. Thus, deriving the diameter from the circumference not only showcases the beauty of geometric relationships but also emphasizes the importance of mathematical literacy in practical scenarios.