Let's start by solving the equation you gave:
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Start with the equation: \[ N \times \sqrt{9} = 12 \] Given that \(\sqrt{9} = 3\), we can rewrite the equation as: \[ N \times 3 = 12 \]
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Solve for \(N\): \[ N = \frac{12}{3} = 4 \]
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Now, we need to establish a formula that connects \(N\) and \(p\). Given that \(p = 25\), we can assume \(N\) might depend linearly on \(p\). Let's look for a linear relationship: \[ N = k \cdot p \] We know that when \(p = 25\), \(N = 4\): \[ 4 = k \cdot 25 \implies k = \frac{4}{25} \] Thus, the formula connecting \(N\) and \(p\) is: \[ N = \frac{4}{25}p \]
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Find \(N\) when \(p = 3\): \[ N = \frac{4}{25} \times 3 = \frac{12}{25} \] Therefore, when \(p = 3\), \(N = \frac{12}{25}\).
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Find \(p\) when \(N = 36\): Using the established formula: \[ 36 = \frac{4}{25} p \] Rearranging gives: \[ p = 36 \times \frac{25}{4} = 9 \times 25 = 225 \] Thus, when \(N = 36\), \(p = 225\).
Summary of Results:
- The formula connecting \(N\) and \(p\) is \(N = \frac{4}{25}p\).
- When \(p = 3\), \(N = \frac{12}{25}\).
- When \(N = 36\), \(p = 225\).