Aim: How do solve related rate problems involving trig functions? Do Now: 1. Evaluate the following: a. sin 𝜋 4 b. tan 𝜋 4 c. sec 𝜋 3 2. Differentiate the following: 𝑎. tan 𝜃 𝑏. sec 𝜃

Mario is standing 25 feet from the launch pad of a hot air balloon Mario is standing 25 feet from the launch pad of a hot air balloon. If the balloon is rising at the rate of 20 ft/min, how will the angle between Mario’s eye and the balloon be changing after 5 minutes? 𝑑𝑦 𝑑𝑡 =20 𝑡𝑎𝑛𝜃= 𝑦 25 z 𝑠𝑒𝑐 2 𝜃 𝑑𝜃 𝑑𝑡 = 1 25 𝑑𝑦 𝑑𝑡 When t = 5, y = 5(20) = 100 y 25 𝜃 𝑠𝑒𝑐𝜃= 103 25 𝑧= 10000+625 ≈ 103 𝑑𝜃 𝑑𝑡 = 1 25 ∙20∙ ( 25 103 ) 2 = 4 5 ∙ 625 10609 =.047 𝑟𝑎𝑑/𝑚𝑖𝑛

A hot-air balloon rising straight up from a level field is tracked by an inclinometer 500 ft from the lift-off point. At the moment the inclinometer’s angle of elevation is 𝜋 4 radians, the angle is increasing at the rate of 0.14 rad/min. How fast is the balloon rising at that moment? 𝑑𝜃 𝑑𝑡 =0.14 𝜃= 𝜋 4 y 500 𝜃 𝑡𝑎𝑛𝜃= 𝑦 500 𝑠𝑒𝑐 2 𝜃 𝑑𝜃 𝑑𝑡 = 1 500 𝑑𝑦 𝑑𝑡 (𝑠𝑒𝑐 𝜋 4 ) 2 ∙0.14= 1 500 𝑑𝑦 𝑑𝑡 𝑑𝑦 𝑑𝑡 =2∙0.14∙500=140 𝑓𝑡/𝑚𝑖𝑛

A camera on the ground 200 meters away from a hot air balloon records the balloon rising into the sky at a constant rate of 10 m/sec. How fast is the camera’s angle of elevation changing when the balloon is 150 meter in the air? 𝑑𝑦 𝑑𝑡 =10 z Finding 𝑑𝜃 𝑑𝑡 y 𝜃 200 𝑠𝑒𝑐 2 𝜃 𝑑𝜃 𝑑𝑡 = 1 200 𝑑𝑦 𝑑𝑡 𝑡𝑎𝑛𝜃= 𝑦 200 Find z before finding 𝑠𝑒𝑐 2 𝜃 𝑧= 40000+22500 =250 250 200 2 𝑑𝜃 𝑑𝑡 = 1 200 ∙10 𝑑𝜃 𝑑𝑡 = 1 20 ∙ 16 25 =.032 𝑟𝑎𝑑/𝑠𝑒𝑐

A plane (alt. 4000 ft) is flying west at 700 ft/s A plane (alt. 4000 ft) is flying west at 700 ft/s. A searchlight, under its path, tracks it. How fast is the light pivoting when the plane is 5000 ft west? 𝑡𝑎𝑛𝜃= 𝑥 4000 x(t) 5000 ft 𝑡𝑎𝑛𝜃= 1 4000 𝑥 𝑠𝑒𝑐 2 𝜃 𝑑𝜃 𝑑𝑡 = 1 4000 𝑑𝑥 𝑑𝑡 4000 𝜃 𝑡𝑎𝑛𝜃= 5000 4000 = 5 4 𝜃=0.896 𝑟𝑎𝑑 𝑠𝑒𝑐 2 .896 𝑑𝜃 𝑑𝑡 = 1 4000 ∙700 2.56 𝑑𝜃 𝑑𝑡 =.175 𝑑𝜃 𝑑𝑡 =.068 𝑟𝑎𝑑𝑖𝑎𝑛 𝑓𝑡/𝑠𝑒𝑐

A camera on the ground 300 meters away from the launchpad records a hot air balloon rising at a rate of 10 m/sec. How fast is the camera’s angle of elevation changing when the hot air balloon is 400 meters high? 𝟑 𝟐𝟓𝟎 𝒓𝒂𝒅/𝒔𝒆𝒄