The woman in the boat is represented by W and the restaurant by R. She rows the distance WP at 3mph and walks the distance PR at 4mph. Let x=BP. WP=√(x²+25) as shown and PR=8-x because BR=8 miles.
Time to row to P=√(x²+25)/3 and time to walk to R from P=(8-x)/4.
Total time T(x)=√(x²+25)/3+(8-x)/4.
T(x) has a turning point at dT/dx=0: differentiate T wrt x: x/(3√(x²+25))-1/4=0.
Cross-multiply: 4x=3√(x²+25), and square: 16x²=9(x²+25)=9x²+225, x²=225/7 so x=15/√7 or 15√7/7.
So the minimum value of T is 20/(3√7)+2-15/(4√7)=(24+5√7)/12=3.1024 hours approx.
We know this is a minimum because if we put x equal to some other value in the interval [0,8], T should be greater than 3. Put x=0: T(0)=5/3+2=11/3 > 3.1. If x=8 (she rows all the way), T(8)=3.1447 > 3.1024.