The sine function computes the area, A, of ABC. B A = - bc sin A 2 C a A = - ac sin B A = - ab sin C 2 A b C Show that bd sin A equals the area of parallelogram = ABCD . D C C b a A d BAn engineer wants to position three pipes at the vertices of a triangle, as shown in the figure below. If the pipes, A, B, and C have radii 2 inches, 3 inches, and 4 inches, respectively, then what are the measures of the angles of triangle ABC. Eavesdroppers use a microphone placed at the focus of a parabolic shield as shown below in order to \"pull in voices\" from far away. sound waves Let the equation x = y2 trace out the parabola of the shield. How far from the shield vertex should the microphone appear to maximize effectiveness? For a satellite in an elliptical orbit around a planet, perigee, P, is the closest distance to the planet and apogee, A, is the greatest distance from the planet. AP A+P' Let 6 represent the eccentricity of an ellipse. Show that e = A ship sails on a hyperbolic trajectory using two different radio stations 202 miles apart as foci. Timing the signals from the stations, 81 and S2, the ship's communication officer determines the ship is 40 miles closer to 51. Write an equation describing the ship's trajectory using the coordinate system in the diagram. The Cauchy-Schwarz Inequality says that if v and w are two vectors, then |vows v w . Under what conditions is vow = vw|?Ohm's Law relates current I , voltage V, and impedance Z with the equation I = Z . Z Use Ohm's Law to nd the current given voltage V = 120(cos 60+i sin 60) and impedance Z = 8 (cos 30 + i sin 30) . Powers of i possess an interesting periodic property. i 21' Thus, every natural number power of i can be expressed as one of the numbers: 1' , 1 , z' , or 1. Use this fact and the pattern above to simplify i228 . Show that a polar equation for the straight line y =mx is 0 =tan"' m.Suppose the parametric equations describe the position of one particle at time t . x1=3sint, y1=2cost, OSISZJI Suppose further that the next pair of parametric equations describe the position of a second particle at time I . x2 =3+cosr, y2 =1+sinr, OStSZr Graphs the paths of both particles and determine the number of literal intersection points in the paths of the particles. Will the particles collide (cross paths at the same time)? A retirement plan gives the recipient a fixed raise of d dollars each year. If the fifth year income was $24,500 and the ninth year income was $25,700 , then what was the first year income? If a patient starts taking 100 mg of a particular drug every 8 hours, then i:100(0.69)k_1 gives k=l the amount of the drug in the patient's bloodstream after taking the nth pill. Calculate the amount of the drug in the patient's bloodstream after 25 pills. Application Exercise 1 Given eight astronauts qualied as pilots and eighteen astronauts qualified as specialists, how many crews with one pilot and two specialists exist? Application Exercise 2 Let P(W) represent the probability of exactly X wins in 11 attempts where the chance to win equals p in every attempt. Then, Only 3 out of 4 patients treating a cardio-vascular condition with surgery survive the surgery. Of 6 of these surgery patients, what is the probability that exactly 5 will survive