The sound intensity 1 decreases with the distance R from the source according to the following law: I=(A)/(R^2) where A is a constant. Find the rate of change of the intensity, Problem 3.

To find the rate of change of the intensity \( I \) with respect to the distance \( R \), we need to differentiate the given equation \( I = \frac{A}{R^2} \) with respect to \( R \).<br /><br />Given:<br />\[ I = \frac{A}{R^2} \]<br /><br />We want to find:<br />\[ \frac{dI}{dR} \]<br /><br />Using the power rule for differentiation, we have:<br />\[ \frac{d}{dR} \left( \frac{A}{R^2} \right) = A \cdot \frac{d}{dR} \left( R^{-2} \right) \]<br /><br />The derivative of \( R^{-2} \) with respect to \( R \) is:<br />\[ \frac{d}{dR} \left( R^{-2} \right) = -2R^{-3} \]<br /><br />So,<br />\[ \frac{dI}{dR} = A \cdot (-2R^{-3}) \]<br />\[ \frac{dI}{dR} = -\frac{2A}{R^3} \]<br /><br />Therefore, the rate of change of the intensity with respect to the distance is:<br />\[ \frac{dI}{dR} = -\frac{2A}{R^3} \]