Problem 51 - Basic Technical Mathematics with Calculus

Question

Using a tape measure, the circumference of a tree is found to be 112 in. What is the diameter of the tree (assuming a circular cross section)?

Step-by-Step Solution

Answer

The diameter of the tree is approximately 35.68 inches.

Step 1: Understand the Relationship Between Diameter and Circumference

The circumference of a circle is related to its diameter by the formula: \( C = \pi \times D \), where \( C \) is the circumference and \( D \) is the diameter.

Step 2: Rearrange the Formula for Diameter

To find the diameter, rearrange the formula to solve for \( D \): \( D = \frac{C}{\pi} \). This allows us to find the diameter when the circumference is known.

Step 3: Substitute the Given Circumference

Substitute the known circumference into the formula. In this case, \( C = 112 \) inches. Thus, \( D = \frac{112}{\pi} \).

Step 4: Calculate the Diameter

To calculate \( D \), use the approximation \( \pi \approx 3.14159 \). Then, \( D = \frac{112}{3.14159} \approx 35.68 \) inches.

Key Concepts

Circumference

Circumference refers to the total distance around the edge of a circle. It’s like measuring the length of a piece of string that would wrap around the circle perfectly. This concept is important in geometry because it helps us understand how big a circle is from the outside.

• In simple terms, it's the perimeter of a circle.
• We typically use the formula to calculate the circumference, which involves the diameter or radius and the constant \( \pi \).

For most practical purposes, knowing the circumference aids in determining other properties, like the diameter if needed. It’s a fundamental concept that forms the basis for more complex problems involving circles.

Diameter

The diameter of a circle is a straight line passing from one side of the circle to the other, through the center. This line cuts the circle into two equal halves and is an essential measurement in circular geometry.

• The diameter is twice the length of the radius, which is the distance from the center of the circle to any point on its edge.
• It gives us a straightforward way to understand the circle's overall size as it spans the entire width of the circle.

Understanding diameter is crucial for computations involving both circumference and area in circular geometry. It directly influences the formula used for circumference, as we’ve seen in the exercise above.

Circle Formula

The circle formula is a set of related mathematic equations that facilitate calculations involving circle measurements, such as radius, diameter, and circumference. The primary formula used in this practice is:

• \( C = \pi \times D \) \,--\, which relates circumference \( C \) with diameter \( D \)
• \( D = \frac{C}{\pi} \) \,--\, allows calculation of diameter when circumference is known
• The constant \( \pi \) (approximately 3.14159) is pivotal in these formulas, representing the ratio of the circumference of any circle to its diameter.

These circle formulas are practical tools in geometry, enabling us to switch between different measurements of a circle precisely and efficiently.