Verified Answer :

  • Start with the equation \(r^2 z - a = 4h\).
  • Add \(a\) to both sides to get \(r^2 z = 4h + a\).
  • Divide by \(r^2\) to isolate \(z\).
  • The final expression is \(z = \boxed{\frac{4h + a}{r^2}}\).

Explanation

  1. Problem Analysis

The problem asks us to solve the equation \(r^2 z - a = 4h\) for the variable \(z\). This is a linear equation in terms of \(z\), where \(r, a,\) and \(h\) are treated as constants. Our objective is to isolate \(z\) on one side of the equation.

  1. Isolating the z-term

To isolate the term involving \(z\), we add \(a\) to both sides of the equation:
\[r^2 z - a + a = 4h + a\]
\[r^2 z = 4h + a\]

  1. Solving for z

Assuming \(r eq 0\), we divide both sides by \(r^2\) to solve for \(z\):
\[\frac{r^2 z}{r^2} = \frac{4h + a}{r^2}\]
\[z = \frac{4h + a}{r^2}\]
This expression gives us the value of \(z\) in terms of the parameters \(r, a,\) and \(h\).

Examples

This formula can be used in physics or engineering problems where you need to isolate a specific variable. For instance, if you are analyzing a circular motion problem, you might have the equation \(r^2 z - a = 4h\), where \(r\) represents the radius, \(z\) could be an unknown angular acceleration, \(a\) is a constant drag factor, and \(h\) is related to height or potential energy. Solving for \(z\) allows you to calculate the angular acceleration based on the given parameters.

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