Problem 12 - Technical Mathematics with Calculus

Question

In a park is a circular fountain whose basin is \(22.5 \mathrm{m}\) in circumference. What is the diameter of the basin?

Step-by-Step Solution

Answer

The diameter of the fountain's basin is approximately \(7.16 \mathrm{m}\).

Step 1: Recall the relationship between circumference and diameter

The circumference (C) of a circle is related to its diameter (d) by the formula: \( C = \pi d \).

Step 2: Solve for the diameter

Using the given circumference \( C = 22.5 \mathrm{m} \), substitute into the formula and solve for the diameter (d): \[ d = \frac{C}{\pi} \].

Step 3: Calculate the diameter

Substitute \( C = 22.5 \mathrm{m} \) and \( \pi \approx 3.14159 \) into the equation: \[ d = \frac{22.5}{3.14159} \approx 7.16 \mathrm{m} \].

Key Concepts

Technical Mathematics

Technical Mathematics is the application of mathematical techniques to solve real-world problems, particularly in technical fields like engineering, computer science, and the natural sciences. It involves using mathematical concepts and tools that are more advanced than the basic arithmetic and algebra found in everyday life.

When solving technical problems, it's essential to have a clear understanding of the mathematical principles at work. The problem involving the circular fountain, for example, requires knowledge of circle geometry and the relationships between key elements like circumference and diameter. Understanding how to manipulate and solve equations is also an integral part of Technical Mathematics, which is evident in the step-by-step solution provided for finding the diameter of the fountain's basin.

Circle Geometry

Circle geometry is a branch of mathematics that focuses on the properties and measurements related to circles. Key concepts include radius, diameter, and circumference.

The diameter of a circle is a straight line passing through the center of the circle, connecting two points on the circumference. It is twice as long as the radius, which extends from the center of the circle to any point on the circumference. The circumference is the distance around the circle, or the perimeter of the circle. These elements are fundamental when working with circle-related problems. For instance, the problem presented involves the circumference of a circular fountain, and to solve for the diameter, an understanding of how these measurements interrelate is crucial.

Circumference Formula

The circumference of a circle is calculated using the formula:
\[ C = \( \)d \]
where \( C \) represents the circumference, \( \( \) \) is the mathematical constant approximately equal to 3.14159, and \( d \) is the diameter of the circle.

This formula is derived from the properties of a circle and is essential for solving problems relating to circular shapes. Understanding the relationship between the diameter and the circumference allows us to determine one measurement when given the other. In the problem concerning the park's circular fountain, knowing the circumference formula is necessary to find the diameter based on the given circumference.

Solving for Diameter

To solve for the diameter of a circle when the circumference is known, we can rearrange the circumference formula:
\[ d = \frac{C}{\( \)} \]

By substituting the given circumference value into the formula, we can isolate and calculate the diameter. For the circular fountain in the park with a circumference of 22.5 meters, the diameter is found by dividing the circumference by \( \( \) \), approximately 3.14159.

Using this method allows us to solve practical problems in real-world contexts, such as determining the size of the fountain's basin. It highlights the practical application of mathematical concepts in everyday situations.