Problem 105 - Prealgebra

Question

The diameter of the "workhorse fleet" of radio telescopes, like the one in Goldstone, California, is 230 feet. What is the circumference of the radio telescope dish to the nearest tenth? Associated Press-Times-Standard 03/09/2010 NASA will repair deep space antenna in California desert.

Step-by-Step Solution

Answer

The circumference is approximately 722.6 feet.

Step 1: Understand the formula for circumference

The circumference of a circle can be found using the formula \( C = \pi \times d \), where \( C \) is the circumference and \( d \) is the diameter. Given that the diameter of the radio telescope is 230 feet, we'll use this formula to find the circumference.

Step 2: Substitute the given diameter into the formula

Substitute the given diameter of 230 feet into the circumference formula: \( C = \pi \times 230 \). This gives us the equation to solve.

Step 3: Calculate the circumference

Use the approximation \( \pi \approx 3.14159 \) for more precision. Thus, compute by multiplying: \( C = 3.14159 \times 230 \). This results in \( C \approx 722.5667 \) feet.

Step 4: Round the result to the nearest tenth

Round the calculated circumference of 722.5667 feet to the nearest tenth. Thus, the circumference is approximately 722.6 feet.

Key Concepts

Understanding Geometry

Geometry is a branch of mathematics that deals with shapes, sizes, and the properties of space. It is all around us, from the layout of your home to the stars in the sky.
In studying geometry, we learn how to measure and understand different forms, including circles, which is what we're focusing on here.

• Shapes: Shapes like circles, triangles, and rectangles have distinct properties and formulas to calculate their dimensions.
• Spaces: Understanding the space that shapes occupy is fundamental in geometry.

A circle, which is a round plane figure whose boundary consists of points equidistant from a fixed center point, plays a central role in both geometry and our exercise on circumference.
To grasp geometry in everyday contexts, think about things like wheels, balls, or plates, which perfectly illustrate how circles look and function.

Circle Measurement Essentials

When it comes to circles, two of the most crucial measurements are the diameter and the circumference. These measurements help define the size and boundary of the circle.
The diameter is a straight line passing through the center of the circle, ending at two points on the boundary. It essentially slices the circle into two equal parts.
Meanwhile, the circumference is the distance around the circle, akin to the perimeter of a polygon. For example, think of a radio telescope dish as a giant circle. Knowing its diameter helps us calculate the circumference, which is essential for understanding its boundary.

• To measure the circumference accurately, one needs to know the diameter first.
• Using these measurements ensures the circle's dimensions can be precisely calculated and applied in practical scenarios.

Understanding these key concepts in circle measurement lays the groundwork for solving many problems involving circles.

Using the Mathematical Formula

To calculate the circumference of a circle, we rely on a fundamental mathematical formula:
\[ C = \pi \times d \]
where \( C \) stands for circumference and \( d \) stands for diameter.
This formula involves \( \pi \), which is a constant approximately equal to 3.14159. It represents the ratio of the circumference of any circle to its diameter.
Using the given diameter of a circle, like the 230 feet of the radio telescope dish, we substitute this value into the formula. This applies to any circle.Here's a quick guide on how to use the formula:

• Identify the known value: the diameter.
• Substitute the diameter value into the formula.
• Multiply \( \pi \) by the diameter to find the circumference.
• Round the result to maintain precision, particularly if dealing with decimal points.

This process not only helps us determine the measurement of the circle's border but also reinforces the concept's universal application in various fields, such as astronomy, construction, and engineering.