a boat on the ocean is 2 mi from the nearest point on a straight shoreline; that point is 13 mi from a…
a boat on the ocean is 2 mi from the nearest point on a straight shoreline; that point is 13 mi from a restaurant on the shore. a woman plans to row the boat straight to a point on the shore and then walk along the shore to the restaurant. complete parts (a) and (b) below. a. if she walks at 3 mi/hr and rows at 2 mi/hr, at which point on the shore should she land to minimize the total travel time? let x be the distance between the nearest point on shore and the point she lands on shore. if t is the time it takes her to get to the restaurant, what is the objective function? t = (type an expression.)
Answer
Explanation:
Step1: Calculate row - time formula
The distance she rows is the hypotenuse of a right - triangle with legs 2 and x. By the Pythagorean theorem, the rowing distance $d_{row}=\sqrt{x^{2}+4}$. Since time $t=\frac{d}{v}$, and the rowing speed $v_{row} = 2$ mi/hr, the time spent rowing $t_{row}=\frac{\sqrt{x^{2}+4}}{2}$.
Step2: Calculate walk - time formula
The distance she walks is $d_{walk}=13 - x$. Since the walking speed $v_{walk}=3$ mi/hr, the time spent walking $t_{walk}=\frac{13 - x}{3}$.
Step3: Form objective function
The total time $T$ is the sum of the rowing time and the walking time. So $T=\frac{\sqrt{x^{2}+4}}{2}+\frac{13 - x}{3}$.
Answer:
$T=\frac{\sqrt{x^{2}+4}}{2}+\frac{13 - x}{3}$