the rowboat is at point R.
the shorted distance point on the shore if point A.
the hours is at point B.
the distance from the rowboat to point A is 4 miles which is the length of line RA.
the distance from point A to the house at point B is 10 miles which is the length of line AB
the distance from point P to the house at point B is x miles which is the length of line PB.
this makes the distance from point A to point P equal to (10 - x).
the distance from the rowboat to point P is equal to the hypotenuse of the right triangle RAP which is the line RP.
that makes the distance equal to sqrt(4^2 + (10-x)^2).
simplify that to get distance from point A to P equal to sqrt(x^2-20x+116).
the basic formula to use is r * t = d
r is the rate.
t is the time.
d is the distance.
the man rows the boat at 3 miles per hour.
the man walks at 5 miles per hour.
let T1 equal the time it takes to get from point R to point P rowing the boat.
let D1 equal the distance from point R to point P.
you get r * t = d becomes 3 * T1 = D1
since D1 = sqrt(x^2-20x+116), you get 3 * T1 = sqrt(x^2-20x+116)
solve for T1 to get T1 = sqrt(x^2-20x+116)/3
let T2 equal the time it takes to get from poinnt P to point B walking.
let D2 equal the distance from point P to point B.
you get r * t = d becomes 5 * T2 = D2
since D2 = x, you get 5 * T2 = x
solve for T2 to get T2 = x/5
let the total time it takes to get from point R to point B through point P equal to T.
you get T = T1 + T2 = sqrt(x^2-20x+116)/3 + x/5
that's your solution.
you can combine that equation into one common denominator to get:
T = (5*sqrt(x^2-20x+116) + 3x)/15
the two equations of T, shown above, are equivalent.
they produce the same value for T.
the domain of the T function is 0 < x < 10, where x represents the distance from point P to point B.
x cannot be 0 because then point P would be the same point as point B.
x cannot be 10 because then point P would be the same point as point A.
that's because the problem statement says that point P is between point A and point B and can therefore not be at either of them.
either of the equations involving T will be your solution, depending on how you want to show them.