Two sociologists have grant money to study school busing in a particular city. They wish to conduct an opinion survey using 687 telephone contacts and 484 house contacts. Survey company A has personnel to do 31 telephone and 11 house contacts per​ hour; survey company B can handle 22 telephone and 16 house contacts per hour. How many hours should be scheduled for each firm to produce exactly the number of contacts​ needed?

Answer:Company A should have 1.35 hours scheduled and company B 29.3 hours scheduled.Step-by-step explanation:This problem can be solved by a system of equationsI am going to say that x is the number of hours that company A should schedule and y is the number of hours that company B should scheduleWe with to handle 687 telephone contacts. Company A can do 31 telephone contacts per hour, and company B can handle 22 telephone contacts per hour.So[tex]31x + 22y = 687[/tex]We with to handle 484 house contacts. Company A can do 11 house contacts per hour, and company B can handle 16 house contacts per hour.So[tex]11x + 16y = 484[/tex]So, we have to solve the following system of equations:[tex]1) 31x + 22y = 687[/tex][tex]2) 11x + 16y = 484[/tex]I am going to write y as a function of x in 1), and replace in 2)[tex]31x + 22y = 687[/tex][tex]22y = 687 - 31x[/tex][tex]y = \frac{687 - 31x}{22}[/tex]Replacing[tex]11x + 16y = 484[/tex][tex]11x + 16\frac{687 - 31x}{22} = 484[/tex]Multiplying everything by 22, we have[tex]242x + 16\frac{687 - 31x} = 10648[/tex][tex]242x + 10992 - 496x = 10648[/tex][tex]254x = 344[/tex][tex]x = 1.35[/tex][tex]y = \frac{687 - 31x}{22} = \frac{687 - 31(1.35)}{22} = 29.3[/tex]Company A should have 1.35 hours scheduled and company B 29.3 hours scheduled.