Problem 51 - Thinking Mathematically

Question

How many plants spaced every 6 inches are needed to surround a circular garden with a 30 -foot radius?

Step-by-Step Solution

Answer

377 plants are needed to surround the garden.

Step 1: Determine Circumference of the Garden

The circumference of a circle is given by the formula \(2\pi r\), where \(r\) is the radius of the circle. In this case, the radius of the garden is 30 feet. Thus the circumference of the garden in feet becomes: \(C = 2\pi (30) = 60\pi\) feet.

Step 2: Convert the Circumference to Inches

As the plant spacing is given in inches, you need to convert the circumference to inches. Remember, 1 foot is equal to 12 inches. So, the circumference of the garden in inches is \(60\pi \times 12 = 720\pi\) inches.

Step 3: Determine the Number of Plants

The plants are to be spaced every 6 inches. So, the total number of plants will be the circumference of the circle divided by the distance between each plant, which is 6 inches. So, the number of plants is \(720\pi / 6 = 120\pi\). This result needs to be rounded to nearest whole number as you can't have fractional plants. Hence, the answer will be 377 plants as \(120\pi\) equals approximately 377.

Key Concepts

Geometry

Geometry is a fascinating field of mathematics that explores the properties and relationships of shapes and spaces. One of the key aspects of geometry is understanding how different shapes work, especially in practical problems. In this exercise, we are dealing with a circular garden. Circles have unique properties, like a constant distance from the center to any point on the edge, known as the radius. Understanding these properties helps when solving problems involving circles, such as calculating the length of their borders or circumference.

Circle Circumference

The circumference of a circle is one of its most important characteristics. It represents the total distance around the circle. To calculate the circumference, we use the formula:

• \( C = 2\pi r \)

where \( r \) is the radius of the circle and \( \pi \) is a constant approximately equal to 3.14159. Understanding this formula is crucial because it provides a direct method to calculate how long it is around a circle based on just one measurement, the radius. When the radius is known, multiplying it by 2 and then by \( \pi \) yields the circle's circumference, essential in various practical applications like planning a garden perimeter.

Unit Conversion

Unit conversion is a critical skill in mathematics and everyday life. It involves changing a measurement from one unit to another, which can be vital in practical situations. In our exercise, we need to compare inches and feet since the plant spacing is in inches, but the radius of the garden is initially given in feet. Understanding that:

• 1 foot = 12 inches

allows us to convert measurements accurately. By multiplying the circumference in feet by 12, we get the measurement in inches, enabling us to determine how many plants are needed at specific intervals. This skill ensures all parts of a problem use consistent units.

Problem Solving Skills

Problem-solving skills are an essential part of working with word problems, especially in mathematics. These skills involve breaking down a problem into smaller, manageable parts. In this exercise, it means:

• Identifying what the problem is asking (the total number of plants needed)
• Using the right mathematical formulas (circumference of a circle)
• Converting units when necessary (feet to inches)
• Performing accurate calculations
• Rounding appropriately

By following a structured approach, we simplify the problem, anticipate possible challenges, and apply the correct operations to find the solution efficiently. Improving these skills can make mathematical challenges more approachable and solvable.