Problem 26 - Elementary Algebra

Question

Use numerical evaluation to evaluate the equations for the following problems. \(C=2 \pi r .\) Find \(C\) if \(\pi\) is approximated by 3.14 and \(r=6\).

Step-by-Step Solution

Answer

Answer: The circumference of the circle is approximately 37.68.

Step 1: Write down the formula for the circumference of a circle

The formula to find the circumference of a circle is given by: \(C = 2\pi r\)

Step 2: Substitute the given values of \(\pi\) and \(r\)

We are given that \(\pi\) is approximated by 3.14 and \(r=6\). We can substitute these values into the formula: \(C = 2 \times 3.14 \times 6\)

Step 3: Perform the numerical evaluation

Now we multiply the numbers to find the value of \(C\): \(C = 2 \times 3.14 \times 6 = 37.68\) So, the circumference of the circle when \(\pi\) is approximated by 3.14 and \(r=6\) is approximately 37.68.

Key Concepts

Circumference of a Circle

The circumference of a circle is essentially the distance around the circle, which can be thought of as the circle's perimeter. For circles, this distance can be calculated using the formula \(C = 2\pi r\), where \(C\) represents the circumference, \(\pi\) (pi) is a constant approximately equal to 3.14159, and \(r\) is the radius of the circle.

In practical scenarios, the precise value of \(\pi\) is often rounded to make calculations simpler, as was done in this exercise where \(\pi\) was approximated to 3.14.

• \(C = 2 \times \pi \times r\) gives us a way to calculate how long the path along the edge of a circle would be.
• Knowing either the circumference or the radius can help us determine the other measurement when \(\pi\) is considered.

Understanding the circumference is crucial in geometry, as it allows us to relate linear distances with circular shapes.

Substitution Method

The substitution method involves directly replacing variables in a formula with specific numerical values given in a problem. This method is incredibly useful in simplifying equations and reaching a solution efficiently, as demonstrated in solving the circumference formula in this exercise.

When you're given an approximation for \(\pi\) (like 3.14 here) and a radius \(r\) (6 in this case), the substitution method allows you to replace \(\pi\) and \(r\) in the circumference equation with these given numbers. This substitutes variables with real-world or problem-specific numbers to make solving practical.

• This method simplifies complex algebraic formulas into manageable arithmetic.
• It's a straightforward approach that enhances problem-solving skills and precision.
• Understanding how and where to substitute makes tackling similar problems easier in the future.

Approximation

Approximation in mathematics is the process of finding values that are close enough to the exact solution for practical purposes, often when an exact number is too complex to use or unnecessary. In this problem, \(\pi\) was approximated as 3.14 to simplify calculations.

Approximation helps in scenarios where we need a quick estimate or when working with measurements that have inherent uncertainty. In mathematical calculations, using approximate values of constants can often streamline processes without significantly compromising accuracy.

• Approximations are crucial when exact values are unnecessary or practically unattainable.
• In calculating the circumference of a circle, using 3.14 instead of 3.14159 should suffice for most practical needs.
• The use of approximations should be balanced between simplicity of calculation and the required precision.

Approximations are especially common in educational settings to help build intuition and understanding without delving into excessive complexity.