91Ó°ÊÓ

A hot-air balloon is rising vertically. From a point on level ground 120 feet from the point directly under the passenger compartment, the angle of elevation to the balloon changes from \(37.1^{\circ}\) to \(62.4^{\circ} .\) How far, to the nearest tenth of a foot, does the balloon rise during this period?

Use the most appropriate method to solve each equation on the interval \([0,2 \pi) .\) Use exact values where possible or give approximate solutions correct to four decimal places. $$ \cos ^{2} x+5 \cos x-1=0 $$

Our cycle of normal breathing takes place every 5 seconds. Velocity of air flow, \(y,\) measured in liters per second, after \(x\) seconds is modeled by $$ y=0.6 \sin \frac{2 \pi}{5} x $$Velocity of air flow is positive when we inhale and negative when we exhale. Within each breathing cycle, when are we exhaling at a rate of 0.3 liter per second? Round to the nearest tenth of a second.

The number of hours of daylight in Boston is given by $$ y=3 \sin \left[\frac{2 \pi}{365}(x-79)\right]+12 $$where \(x\) is the number of days after January \(l\). Within a year, when does Boston have 10.5 hours of daylight? Give your answer in days after January 1 and round to the nearest day.

When throwing can object, the distance achieved depends on its initial velocity, \(v_{0}\) cand the angle above the horizontal at which the object is thrown, \(\boldsymbol{\theta}\). The distance, \(d,\) in feet, that describes the range covered is given by $$ d=\frac{v_{0}^{2}}{16} \sin \theta \cos \theta $$$$ \text { where } \boldsymbol{\tau}_{0} \text { is measured in feet per second.} $$ You and your friend are throwing a baseball back and forth. If you throw the ball with an initial velocity of \(x_{0}=90\) feet per second, at what angle of elevation, \(\theta\). to the nearest degree, should you direct your throw so that it can be easily caught by your friend located 170 feet away?