13. \( P \propto \sqrt{y} \) and \( P=77 \) when \( y=121 \). Find the law connecting \( P \) and \( y \). Find \( P \) when \( y=144 \). Also, find \( y \) when \( P=22 \). 14. \( O \) varies inversely as \( z \). If \( Q=7 \) when \( z=15 \), find

To solve the problems, we will use the relationships given in each case. ### Problem 13 1. **Finding the law connecting \( P \) and \( y \)**: Given that \( P \propto \sqrt{y} \), we can express this as: \[ P = k \sqrt{y} \] where \( k \) is a constant. We know that \( P = 77 \) when \( y = 121 \). We can use this information to find \( k \): \[ 77 = k \sqrt{121} \] Since \( \sqrt{121} = 11 \), we have: \[ 77 = k \cdot 11 \] Solving for \( k \): \[ k = \frac{77}{11} = 7 \] Therefore, the law connecting \( P \) and \( y \) is: \[ P = 7 \sqrt{y} \] 2. **Finding \( P \) when \( y = 144 \)**: We substitute \( y = 144 \) into the equation: \[ P = 7 \sqrt{144} \] Since \( \sqrt{144} = 12 \): \[ P = 7 \cdot 12 = 84 \] 3. **Finding \( y \) when \( P = 22 \)**: We set \( P = 22 \) in the equation: \[ 22 = 7 \sqrt{y} \] Solving for \( \sqrt{y} \): \[ \sqrt{y} = \frac{22}{7} \] Squaring both sides to find \( y \): \[ y = \left(\frac{22}{7}\right)^2 = \frac{484}{49} \] ### Summary for Problem 13: - The law connecting \( P \) and \( y \) is \( P = 7 \sqrt{y} \). - When \( y = 144 \), \( P = 84 \). - When \( P = 22 \), \( y = \frac{484}{49} \). --- ### Problem 14 1. **Finding the relationship for \( O \)**: Given that \( O \) varies inversely as \( z \), we can express this as: \[ O = \frac{k}{z} \] where \( k \) is a constant. We know that \( Q = 7 \) when \( z = 15 \). We can use this information to find \( k \): \[ O = 7 \quad \text{when} \quad z = 15 \] Thus: \[ 7 = \frac{k}{15} \] Solving for \( k \): \[ k = 7 \cdot 15 = 105 \] Therefore, the relationship is: \[ O = \frac{105}{z} \] ### Summary for Problem 14: - The relationship is \( O = \frac{105}{z} \). If you have any further questions or need additional calculations, feel free to ask!