Problem 71 - Prealgebra

Question

Marta has 61 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 is approximately 19.43 feet.

Step 1: Understand the Problem

The problem provides the total length of the decorative fencing Marta has, which is 61 feet. This fencing will create a circular garden. We are to find the diameter of this circle.

Step 2: Recall the Formula for Circumference

The circumference of a circle is given by the formula \( C = \pi d \), where \( C \) is the circumference and \( d \) is the diameter.

Step 3: Substitute and Solve for Diameter

We know the circumference (\( C \)) is 61 feet, and \( \pi \) is approximately 3.14. Substitute into the formula and solve for \( d \): \( 61 = 3.14d \).

Step 4: Calculate the Diameter

To find \( d \), divide both sides of the equation \( 61 = 3.14d \) by 3.14. This gives \( d = \frac{61}{3.14} \approx 19.43 \).

Step 5: Provide the Answer

The diameter of the circular garden is approximately 19.43 feet when rounded to the nearest hundredth.

Key Concepts

Diameter

The diameter of a circle is an important concept in geometry. It is the longest distance across the circle, passing through its center. This means that if you were to draw a line from one side of the circle straight through the center to the other side, you would have drawn the diameter. In simple terms:

• The diameter is twice the radius, which is the distance from the center of the circle to any point on its edge.
• The formula for the diameter when you have the circumference is: \( d = \frac{C}{\pi} \).
• The diameter is a linear measurement, typically given in units like feet, meters, or inches.

In Marta's case, the diameter of her garden helps her understand the size of the circular area she will enclose with the 61 feet of fencing. Knowing how to find the diameter from the circumference is handy for all sorts of practical problems involving circles, from garden layout to architectural design.

Circle

A circle is a perfectly round shape. Every point on its boundary is equidistant from the center point. This unique property makes circles fascinating in geometry. Here are a few key points about circles:

• All circles are defined by their radius or diameter, and their entire edge is known as the circumference.
• Every circle has a center point, which is the same distance from any point on the circle's edge.
• The formula for the area of a circle is \( A = \pi r^2 \), though not needed here, it helps when discussing properties of circles.

In the garden problem, using the circle as the shape helps maximize the area Marta encloses. Circles are often chosen for gardens because they have a balanced look and feel. The circle's properties simplify calculations and design plans for projects.

Geometry

Geometry is a branch of mathematics that deals with shapes, sizes, and their properties. It is the study of everything from points and lines to complex 3D shapes. Understanding geometry helps us solve real-world problems involving space and form. Here’s how geometry is relevant to Marta’s problem:

• Geometry provides the formula to find the circumference of a circle using its diameter or radius.
• By rearranging and working with these formulas, we can solve for unknowns, like the diameter in Marta’s fencing problem.
• The geometric principle that all points on a circle are equidistant from the center aids in creating designs that are symmetric and balanced.

Geometry not only helps in solving equations but also aids in visualizing and designing spaces, making it invaluable for both practical applications and theoretical exploration. Through understanding geometry, Marta can efficiently plan and execute her garden design, ensuring she uses her resources optimally.