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Chapter 2: Problem 21

Find the value of the remaining variable in each formula. Use 3.14 as an approximation for \(\pi(p i) .\) \(C=2 \pi r\) (circumference of a circle); \(C=16.328\)

Short Answer

Expert verified
The radius \( r \) is approximately 2.6 units.

Step by step solution

01

Write down the given formula

The formula given is the circumference of a circle, which is \(C = 2 \pi r\).
02

Substitute the known value

We know that the circumference \(C = 16.328\). Substitute this into the formula to get: \(16.328 = 2 \pi r\).
03

Use the approximation for \( \pi \)

Approximately, \( \pi = 3.14 \). Substitute this value into the equation: \(16.328 = 2 \times 3.14 \times r\).
04

Solve for the radius \( r \)

Rearrange the equation to solve for \( r \): \( r = \frac{16.328}{2 \times 3.14}\).Perform the calculation: \( r = \frac{16.328}{6.28}\approx 2.6 \text{ units} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

circle circumference
The circumference of a circle is the distance around the circle. Imagine it as the perimeter for circles. The formula used to calculate circumference is \( C = 2 \pi r \). Here, \( C \) is the circumference and \( r \) is the radius of the circle. The symbol \( \pi \) (pi) is a special number used in circle calculations. Using this formula, you can find the circumference if you know the radius, or vice versa.
solving equations for a variable
When you have an equation, you can solve for an unknown variable by isolating it. For example, if you know the circumference \( C \) and need to find the radius \( r \) in the formula \( C = 2 \pi r \), follow these steps:
  • First, write down the known values and substitute them into the equation.
  • Next, rearrange the formula to isolate the unknown variable (radius \( r \)).
    • For example, if \( C = 16.328 \), substitute this into the formula to get \( 16.328 = 2 \pi r \). Then rearrange it to find \( r \): \( r = \frac{16.328}{2 \pi } \). This step-by-step approach helps simplify the problem and find the solution.
approximation of pi (蟺)
Pi (\( \pi \)) is an irrational number often approximated as 3.14. In many math problems, using 3.14 makes calculations simpler and easier. For example, in our problem, we used \( \pi = 3.14 \) to find the radius. So, the equation \( 16.328 = 2 \pi r \) becomes \( 16.328 = 2 \times 3.14 \times r \). This makes it easier to solve real-life problems with a reasonable level of accuracy.
radius calculation
The radius of a circle is the distance from the center of the circle to any point on its edge. Finding the radius involves solving the equation \( C = 2 \pi r \). If we know the circumference and use the approximate value of \( \pi \), we can calculate the radius. For instance, given \( C = 16.328 \), we substitute into the formula and rearrange it:
\( r = \frac{16.328}{2 \times 3.14} \)
Performing the arithmetic, \( r = \frac{16.328}{6.28} \approx 2.6 \). This results in the radius being approximately 2.6 units.

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