Question

a small island is 5 miles from the nearest point p on the straight shoreline of a large lake. if a woman on the island can row a boat 2 miles per hour and can walk 3 miles per hour, where should the boat be landed in order to arrive at a town 10 miles down the shore from p in the least time? let x be the distance (in miles) between point p and where the boat lands on the lakeshore.
(a) enter a function t(x) that describes the total amount of time the trip takes as a function of the distance x
t(x) =
(b) what is the distance x = c that minimizes the travel time?
c =
(c) what is the least travel time?
the least travel time is

Explanation:

Step1: Calculate rowing distance and time

The rowing distance is $\sqrt{x^{2}+5^{2}}$ (by Pythagorean theorem). Rowing speed is $2$ miles per hour. So rowing time $t_{1}=\frac{\sqrt{x^{2}+25}}{2}$.

Step2: Calculate walking distance and time

The walking distance is $10 - x$. Walking speed is $3$ miles per hour. So walking time $t_{2}=\frac{10 - x}{3}$.

Step3: Form the total - time function

The total - time function $T(x)=t_{1}+t_{2}=\frac{\sqrt{x^{2}+25}}{2}+\frac{10 - x}{3}$.

Answer:

$T(x)=\frac{\sqrt{x^{2}+25}}{2}+\frac{10 - x}{3}$