n a laboratory experiment, an ultrasound detector located at D is used to measure the distances of two moving objects, P and Q. At a certain moment, P and Q are located as shown in the diagram. What is distance PQ?

Triangle PDQ has: 

     Side "p" is opposite Angle P 

     Side "d" is opposite Angle D = 45º + 23º = 68º 

     Side "q" is opposite Angle Q 


Use the Law of Cosines: 

         d²  =  p²  +  q²   −   2 • p • q • cos(D) 

         d²  =  (4.24)²  +  (4.24)²   −   2 • (4.24) • (4.24) • cos(68º) 

         d²  =  2(4.24)²   −   2(4.24)² • cos(68º) 

         d²  =  2(4.24)² • [ 1 – cos(68º) ] 

         d  =  4.74 ft
Hope this helps