What is the answer to this problem?

Question

A man in a rowboat that is a = 2 miles from the nearest point A on a straight shoreline wishes to reach a house located at a point B that is b = 8 miles farther down the shoreline (see the figure). He plans to row to a point P that is between A and B and is x miles from the house, and then he will walk the remainder of the distance. Suppose he can row at a rate of 2 mi/hr and can walk at a rate of 5 mi/hr. If T is the total time required to reach the house, express T as a function of x. 

T(x) =  

 

 

Answer 1

First figure out the distance from the boat to point P. Note this is the hypotenuse to the right triangle. A=2, and the base leg = (8-x), so using the Pythagorean theorem, you would get:

c² = (8-x)² + 4²

c² = 64 - 16x + x² +16

c^{2 =} x² - 16x + 80

c = √(x² - 16x + 80) length of the diagonal leg (at speed 2mph)

distance from there to the house is simply x (and speed is 5mph) so....

T(x) = (2)(√(x² - 16x + 80)) + 5(x)