Ava and Kelly ran a road race, starting from the same place at the same time. Ava ran at an average speed of 6 miles per hour. Kelly ran at an average speed of 8 miles per hour.When will Ava and Kelly be 3/4 mile apart?

Hello!The answer is:Ava and Kelly will be 3/4 mile apart after 0.375 hours.Why?To calculate when will Ava and Kelly be 3/4 mile apart, we need to write the equations for both Ava's and Kelly's positions.Writing the equations we have:For Ava:[tex]x_A=x_o+v_ot\\\\x_A=0+v_o*t\\\\x_A=v_{o(ava)*t[/tex]For Kelly:We need to calculate when Kelly will be 3/4 mile apart becase is running faster than Ava, so, writing the equation we have:[tex]x_K=x_A+\frac{3}{4}mile=x_o+v_{o(Kelly)}*t[/tex]Now, substituting the equation for Ava into the equation for Kelly, we have:[tex]x_A+\frac{3}{4}mile=x_o+v_{o(Kelly)}*t[/tex][tex]v_{o(ava)*t+\frac{3}{4}mile}=x_o+v_{o(Kelly)}*t[/tex][tex]v_{o(ava)*t+0.75mile}=x_{o}+v_{o(Kelly)}*t\\\\6mph*t+0.75mile=8mph*t\\\\0.75mile=2mph*t\\\\t=\frac{0.75miles}{2mph}=0.375hours[/tex]To prove  that the result is correct, we just need to substitute the obtained value for time into both equations, so, substutiting we have:For Ava:[tex]x_A=v_{o(ava)*t=6mph*0.375hours=2.25miles[/tex]For Kelly:[tex]x_K=v_{o(Kelly)}*t=8mph*0.375=3miles[/tex]There is a difference of 0.75 miles or 3/4 mile between Ava and Kelly, so, the obtained value for time is correct.Therefore, we have that Ava and Kelly will be 3/4 mile apart after 0.375 hours.Have a nice day!