Question 1145867
my understanding of this problem is shown in the following diagram.


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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.