Problem 9 - Algebra A Combined Function

Question

Substitute the given values into each given formula and solve for the unknown variable. $$ C=2 \pi r ; \quad C=15.7 $$

Step-by-Step Solution

Answer

The radius \( r \) is approximately 2.5.

Step 1: Identify the Given Values

We are given that the formula is \( C = 2 \pi r \) and the value of \( C \) is 15.7. The task is to substitute \( C = 15.7 \) and solve for \( r \) (the radius).

Step 2: Substitute into the Formula

Begin by substituting the given value of \( C \) into the formula: \[ 2 \pi r = 15.7 \] This equation needs to be solved for \( r \).

Step 3: Isolate the Variable

To solve for \( r \), divide both sides of the equation by \( 2 \pi \):\[ r = \frac{15.7}{2 \pi} \]This isolates \( r \) on one side of the equation.

Step 4: Calculate the Value of the Radius

Now calculate the value of \( r \):\[ r \approx \frac{15.7}{6.2832} \]Using the approximate value \( \pi \approx 3.1416 \), the calculation yields:\[ r \approx 2.5 \] Thus, \( r \) is approximately 2.5.

Key Concepts

Formula Manipulation

Formula manipulation is a technique used to rearrange formulas to solve for a particular variable. This is especially useful in algebra when dealing with equations involving multiple variables. The goal is to isolate the variable of interest on one side of the equation, making it easier to solve.

To begin with, identify the variable you need to solve for. Then, use inverse operations to move other terms to the opposite side of the equation. In the case of the formula for circumference,

• we identified that we needed to solve for the radius \( r \),
• with the original formula being \( C = 2 \pi r \).

Substitute the known value, which in this case was the circumference \( C = 15.7 \), into the formula, then rearrange it so that the variable \( r \) is isolated on one side of the equation. This allows the calculation of \( r \) by simplifying the formula to \( r = \frac{15.7}{2\pi} \).Using formula manipulation not only aids in calculations but also enhances understanding by breaking down complex equations into manageable steps.

Solving Equations

Solving equations involves finding the values of variables that make an equation true. This is a fundamental concept in algebra.

• Start by rewriting the equation so that it is easy to identify what needs to be solved.
• The goal is to isolate the unknown variable on one side, which involves performing inverse operations such as addition, subtraction, multiplication, or division on both sides of the equation.

In our exercise, after substituting the value \( C = 15.7 \) into the formula \( C = 2 \pi r \), we needed to solve for \( r \). Thus, division was used as an inverse operation to isolate \( r \) by dividing both sides by \( 2 \pi \), leading to the equation \( r = \frac{15.7}{2 \pi} \).

Once simplified, the approximate value of the variable can be calculated using a known or estimated value of constants, such as \( \pi \approx 3.1416 \). This yields the approximate solution \( r \approx 2.5 \). Understanding solving equations allows you to tackle a variety of math problems efficiently.

Circumference Calculation

Circumference calculation is essential when dealing with circles in math. The circumference \( C \) refers to the distance around the circle, and it can be calculated using the formula:

• \( C = 2 \pi r \), where \( r \) represents the radius,
• \( \pi \) is a mathematical constant approximately equal to 3.1416.

When you have the value of the circumference and need to find the radius, you will reverse the calculation process. Substitute the circumference value into the formula, as seen in our example with \( C = 15.7 \), then solve the equation for \( r \) by rearranging to \( r = \frac{15.7}{2 \pi} \).

Such calculations not only serve practical applications, like measuring around circular objects, but also reinforce understanding of how different geometric measures are interconnected. Mastery of circumference calculations aids in comprehending larger mathematical concepts and geometry principles.