Problem 66 - Prealgebra

Question

Trinity has 44 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 14.01 feet.

Step 1: Understand the Formula for Circumference

The circumference of a circle (the distance around the circle) is given by the formula \( C = \pi \times d \), where \( d \) is the diameter and \( \pi \) is approximately 3.14 for this problem. Trinity's decorative fencing measures 44 feet, which serves as the circumference of the circular garden.

Step 2: Set Up the Equation

We know that \( C = 44 \) feet. Plugging the known values into the circumference formula, we get the equation \( 44 = 3.14 \times d \) where \( d \) is the diameter we need to find.

Step 3: Solve for the Diameter

To find the diameter \( d \), we need to isolate \( d \) in the equation \( 44 = 3.14 \times d \). We do this by dividing both sides of the equation by 3.14: \( d = \frac{44}{3.14} \).

Step 4: Calculate the Diameter

Perform the division \( \frac{44}{3.14} \) to find \( d \). Using a calculator, this yields \( d \approx 14.01 \).

Step 5: Round the Diameter

The calculation gives us \( d \approx 14.0127 \). Rounding to the nearest hundredth, we get \( d \approx 14.01 \). Thus, the diameter of the circular garden is 14.01 feet.

Key Concepts

Diameter

The diameter of a circle is a straight line that passes from one side of the circle to the other, going through the center point. It is one of the most important measurements in a circle as it directly relates to the circumference and the area.

To find the diameter of Trinity's circular garden, we know the circumference (C) is 44 feet. The formula for circumference involves the diameter: \[ C = \pi \times d \]In our specific exercise, this formula becomes \[ 44 = 3.14 \times d \] Solving this equation involves dividing both sides by \( \pi \): \[ d = \frac{44}{3.14} \]Using a calculator, this results in a diameter of approximately 14.01 feet. This statement signifies that the garden's diameter stretches 14.01 feet across, cutting the circle into two equal halves.

Pi (π)

Pi (π) is a special number that originates from mathematics. It is critical when dealing with circles because it helps in finding both the circumference and area. The exact value of \( \pi \) is an irrational number, which means it cannot be precisely written down in decimals, but it is commonly approximated as 3.14 for ease of calculation.

In Trinity's garden exercise, \( \pi \) serves as a multiplier to determine the circumference from the diameter. The relationship is expressed in the formula: \[ C = \pi \times d \]This formula unfolds the circle's perimeter by scaling the diameter with \( \pi \) producing an expansive and accurate representation of the circle's boundary.

Circular Garden

A circular garden, like the one Trinity plans to build, is characterized by its geometric shape - a circle. Circles have symmetrical properties making them aesthetically pleasing and easy to measure when you know a few vital parts, like the radius, the diameter, or the circumference.

The circumference is particularly useful when you're dealing with the perimeter of the circle, such as when fencing it. In the given problem, Trinity has 44 feet of fencing available to enclose her garden. This length corresponds to the circle's circumference, giving us clues to calculate other dimensions like the diameter.

• The garden's diameter represents the largest cross-section possible within the circle.
• This structure portrays natural harmony because of its equidistant points from the center.

Such properties make circles ideal for gardens, ensuring a balanced distribution of elements and an effortless design even in spatially constrained areas.