Questions: Maximizing the Viewing Angle A billboard that is 30 ft wide is placed 60 ft from a highway as shown in the figure. The billboard is easiest to read from the highway when the viewing angle α is larger than 7°. a) Show that α = tan⁻¹(90 / x) - tan⁻¹(60 / x). b) Graph the function in part (a). c) For what approximate values of x is α greater than 7°? d) For approximately how long does a motorist traveling at 35 mph have a viewing angle larger than 7°?

a) Show that $\alpha=\tan ^{-1}(90 / x)-\tan ^{-1}(60 / x)$. Find the angle from the car to the far end of the billboard. Let $\beta$ be the angle from the car to the far end of the billboard. Then $\tan(\beta) = \frac{60+30}{x} = \frac{90}{x}$, so $\beta = \tan^{-1}(\frac{90}{x})$. Find the angle from the car to the near end of the billboard. Let $\gamma$ be the angle from the car to the near end of the billboard. Then $\tan(\gamma) = \frac{60}{x}$, so $\gamma = \tan^{-1}(\frac{60}{x})$. Express $\alpha$ in terms of $\beta$ and $\gamma$. $\alpha = \beta - \gamma = \tan^{-1}(\frac{90}{x}) - \tan^{-1}(\frac{60}{x})$. $\boxed{\alpha = \tan^{-1}(\frac{90}{x}) - \tan^{-1}(\frac{60}{x})}$

b) Graph the function in part (a). Define the function. $f(x) = \tan^{-1}(\frac{90}{x}) - \tan^{-1}(\frac{60}{x})$. Plot the function. The graph of the function starts at 0 when $x$ is close to 0 and increases to a maximum, then decreases and approaches 0 as $x$ goes to infinity. The maximum viewing angle is approximately $11.86^\circ$ when $x \approx 74$. $\boxed{\text{See a graphing calculator or software for the graph.}}$

c) For what approximate values of $x$ is $\alpha$ greater than $7^{\circ}$? Set up the inequality. $\tan^{-1}(\frac{90}{x}) - \tan^{-1}(\frac{60}{x}) > 7^{\circ} = 7 \cdot \frac{\pi}{180}$ radians $\approx 0.122$ radians. Solve the inequality graphically. Using a graphing calculator or software, find the values of $x$ where the graph of $\alpha(x)$ is above $7^{\circ}$. Determine the range of $x$. The approximate range of $x$ where $\alpha > 7^{\circ}$ is from about $x = 35$ to $x = 205$. $\boxed{35 < x < 205}$

d) For approximately how long does a motorist traveling at 35 mph have a viewing angle larger than $7^{\circ}$ ? Calculate the distance the motorist travels with a viewing angle greater than $7^{\circ}$. The motorist travels a distance of $205 - 35 = 170$ ft while the viewing angle is greater than $7^{\circ}$. Convert the speed from mph to ft/s. $35 \text{ mph} = 35 \cdot \frac{5280 \text{ ft}}{3600 \text{ s}} \approx 51.33 \text{ ft/s}$. Calculate the time the motorist has a viewing angle greater than $7^{\circ}$. Time $= \frac{\text{distance}}{\text{speed}} = \frac{170 \text{ ft}}{51.33 \text{ ft/s}} \approx 3.31$ seconds. $\boxed{3.31 \text{ seconds}}$

$\alpha = \tan^{-1}(\frac{90}{x}) - \tan^{-1}(\frac{60}{x})$ See a graphing calculator or software for the graph. $35 < x < 205$ $3.31$ seconds