Verified Answer :

  • Start with the given equation \(\frac{1}{4} b r = j\).
  • Multiply both sides by \(4\) to remove the fraction, resulting in \(b r = 4 j\).
  • Divide both sides by \(r\) (assuming \(r eq 0\)) to isolate \(b\).
  • The solution is \(b = \boxed{\frac{4j}{r}}\).

Explanation

  1. Problem Analysis

The goal is to isolate the variable \(b\) in the given equation \(\frac{1}{4} b r = j\). Here, we consider \(r\) and \(j\) as constants, and we assume \(r eq 0\) to ensure the division is well-defined. Our objective is to manipulate the equation algebraically to express \(b\) in terms of \(j\) and \(r\).

  1. Eliminating the Fraction

Starting with the equation \[\frac{1}{4} b r = j\] we first want to eliminate the fraction. To do this, multiply both sides of the equation by 4: \[\left(4\right) \times \left(\frac{1}{4} b r\right) = 4 \times j\] which simplifies to \[b r = 4 j\]

  1. Isolating \(b\)}}],example_of_usage:

Now that we have the equation in the form \(b r = 4 j\), we isolate \(b\) by dividing both sides by \(r\). Since we assume \(r eq 0\), we can safely perform this division: \[b = \frac{4 j}{r}\] This is the final expression for \(b\).

  1. Real-world Application Example

Consider a scenario where you are calculating the base length of a triangle if you know the area and the height. The area of a triangle is given by the formula \(A = \frac{1}{2} b h\), where \(b\) is the base and \(h\) is the height. If you were given a different geometric relationship, say the area is \(\frac{1}{4}\) of the product of the base and some parameter \(r\), i.e., \(A = \frac{1}{4} b r\), you can use the same algebraic steps demonstrated here to find the length of the base as \(b = \frac{4A}{r}\). This technique of isolating a variable is fundamental in physics and engineering for rearranging formulas to solve for unknowns in various experimental contexts.

Examples

Consider a scenario where you are calculating the base length of a triangle if you know the area and the height. The area of a triangle is given by the formula \(A = \frac{1}{2} b h\), where \(b\) is the base and \(h\) is the height. If you were given a different geometric relationship, say the area is \(\frac{1}{4}\) of the product of the base and some parameter \(r\), i.e., \(A = \frac{1}{4} b r\), you can use the same algebraic steps demonstrated here to find the length of the base as \(b = \frac{4A}{r}\). This technique of isolating a variable is fundamental in physics and engineering for rearranging formulas to solve for unknowns in various experimental contexts.

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