A surveyor standing 50 feet from the base of a large tree measures the angle of elevation to the top of the trie as \( 65.3^{\circ} \). How accurately must the angle be measured if the percent errar in estimating the height of the tree is to be less than \( 3 \% \) ? (Round your answer to three decimal places.) \( |d \theta| \leq \square \) radians Need Help? Readit

Let the height of the tree be given by \[ h = 50 \tan \theta, \] where \(\theta = 65.3^\circ\). We wish to know how accurately the angle must be measured so that the percent error in \(h\) is less than \(3\%\). Step 1. Differentiate \(h\) with respect to \(\theta\). \[ \frac{dh}{d\theta} = 50 \sec^2 \theta. \] Step 2. Using linear approximation for small errors, a small error \(d\theta\) in \(\theta\) produces an error in \(h\) of \[ dh \approx 50 \sec^2 \theta\, d\theta. \] Step 3. The relative error in \(h\) is approximately \[ \frac{dh}{h} \approx \frac{50 \sec^2 \theta\, d\theta}{50 \tan \theta} = \frac{\sec^2 \theta}{\tan \theta}\, d\theta. \] Step 4. We require that \[ \left|\frac{dh}{h}\right| \leq 0.03. \] Thus, \[ \left|\frac{\sec^2 \theta}{\tan \theta}\, d\theta\right| \leq 0.03. \] Step 5. Solving for \(|d\theta|\) gives \[ |d\theta| \leq 0.03\, \frac{\tan \theta}{\sec^2 \theta}. \] Step 6. Now, substitute \(\theta = 65.3^\circ\). We first compute: \[ \tan 65.3^\circ \approx 2.174, \quad \text{and} \quad \sec^2 65.3^\circ = 1 + \tan^2 65.3^\circ \approx 1 + (2.174)^2 \approx 1 + 4.727 \approx 5.727. \] Step 7. Substitute these values: \[ |d\theta| \leq 0.03\, \frac{2.174}{5.727} \approx 0.03 \times 0.380 \approx 0.0114 \text{ radians}. \] Rounded to three decimal places, the required accuracy is \[ |d\theta| \leq 0.011 \text{ radians}. \]