Answer:
4.2 mph
Explanation:
First, recall the formula below:
[tex]Distance=Speed\times Time[/tex]Let the number of hours for which they walked = t
Starting from the pretzel stand, Jazmine leaves walking due East toward Foot Locker at 3 mph.
[tex]\text{ Distance covered by Jasmine after t hours}=3t\text{ miles}[/tex]James 3 minutes later leaves walking due North toward Dillard's at 3 miles per hour.
[tex]\text{ Distance covered by James after \lparen}t-0.05)\text{ hours}=3(t-0.05)\text{ miles}[/tex]The diagram below illustrates the given information:
Using the Pythagorean theorem, we have that:
[tex]x^2=(3t)^2+[3(t-0.05)]^2[/tex]At t=12 minutes = 12/60 = 0.2 hours
[tex]\begin{gathered} x^2=0.5625 \\ \implies x=\sqrt{0.5625}=0.75\text{ miles} \end{gathered}[/tex]The distance, x between the two after 12 minutes is 0.75 miles.
Next, take the derivative of the equation:
[tex]\begin{gathered} x^2=(3t)^2+[3(t-0.05)]^2 \\ Simplify \\ x^2=18t^2-0.9t+0.0225 \\ Take\;the\;derivative \\ 2x\frac{dx}{dt}=36t-0.9 \\ \implies\frac{dx}{dt}=\frac{36t-0.9}{2x} \end{gathered}[/tex]At t=0.2 hours, x=0.75 miles
[tex]\frac{dx}{dt}=\frac{36(0.2)-0.9}{2(0.75)}=\frac{6.3}{1.5}=4.2\text{ miles per hour}[/tex]The rate at which the distance between them is changing 12 minutes after is 4.2 miles per hour.
