Problem 29 - College Algebra

Question

The formula \(F=\frac{m v^{2}}{R}\) relates force \((F),\) velocity \((v)\) \(\operatorname{mass}(m),\) and resistance \((R) .\) Find \(R\) when \(m=45\), \(v=7,\) and \(F=245\)

Step-by-Step Solution

Answer

The resistance \( R \) is 9.

Step 1: Understand the Given Formula

The given formula is meant to calculate the force \( F \), as a function of mass \( m \), velocity \( v \), and resistance \( R \). It is written as \( F = \frac{m v^2}{R} \). In this problem, we want to find \( R \).

Step 2: Identify Known Values

From the exercise, we know the following values: mass \( m = 45 \), velocity \( v = 7 \), and force \( F = 245 \).

Step 3: Rearrange the Formula to Solve for \( R \)

To solve for resistance \( R \), rearrange the formula to isolate \( R \) on one side. Start by multiplying both sides by \( R \) to get rid of the denominator:\[ F \, R = m v^2 \]Now, solve for \( R \) by dividing both sides by \( F \):\[ R = \frac{m v^2}{F} \]

Step 4: Substitute the Known Values into the Equation

Now substitute the known values into the rearranged formula:\[ R = \frac{45 \times 7^2}{245} \].

Step 5: Calculate \( v^2 \)

Calculate the square of the velocity:\[ v^2 = 7^2 = 49 \]

Step 6: Plug \( v^2 \) into the Equation

Substitute \( v^2 = 49 \) back into the equation for \( R \):\[ R = \frac{45 \times 49}{245} \]

Step 7: Simplify the Equation to Find \( R \)

Calculate the expression:\[ 45 \times 49 = 2205 \]Now divide by 245:\[ R = \frac{2205}{245} = 9 \]

Step 8: Conclusion

The resistance \( R \) is calculated to be 9.

Key Concepts

Formula Rearrangement

In algebra, it's often necessary to rearrange formulas to isolate and solve for a particular variable. This involves changing the structure of the equation so that the desired variable stands alone on one side, with everything else on the other side. In the context of our problem, we start with the formula for force: - Given: \(F = \frac{mv^2}{R}\)

• We need \(R\) to be isolated, so first, multiply both sides by \(R\) to eliminate the fraction.
• This gives us: \(FR = mv^2\).
• Next, divide both sides by \(F\) to solve for \(R\), resulting in: \(R = \frac{mv^2}{F}\).

Rearranging formulas like this helps us express relationships between quantities and makes it easier to substitute known values in later steps.

Mathematical Calculation

Solving mathematical problems often involves calculating values step by step to simplify the expression and find the answer. In our exercise on finding resistance \(R\), this process involves multiple calculations: - Start with the rearranged formula: \(R = \frac{mv^2}{F}\).

• Calculate \(v^2\) first, since this value directly affects the numerator of our fraction. For \(v = 7\), \(v^2 = 49\).
• Substitute \(m = 45\), \(v^2 = 49\), and \(F = 245\) into the equation: \(R = \frac{45 \times 49}{245}\).
• Perform the multiplication to get \(45 \times 49 = 2205\).
• Finally, divide 2205 by 245 to find \(R = 9\).

Every step involves careful computation, ensuring that we follow correct mathematical principles to yield the right answer.

Problem-Solving Steps

Problem-solving in mathematics typically follows a series of organized steps that lead to a solution. This structured approach not only helps in an orderly execution but also in understanding the problem deeply. For our exercise, the steps include:- **Understand the Problem:** Identify the unknown we need to solve, which in this case, is \(R\).- **List Known Values:** Gather all given information, such as \(m = 45\), \(v = 7\), \(F = 245\).- **Rearrange the Equation:** Transform the equation to isolate the unknown, forming \(R = \frac{mv^2}{F}\).

• Substitute the known values into the rearranged formula.
• Carry out the necessary mathematical operations—squaring, multiplying, dividing—to simplify the equation.
• Conclude with the solution, verifying that the calculations are correct.

Following these steps ensures that each part of the problem is addressed in order, reducing confusion and setting a clear path to finding the solution.