A boat on the ocean is 4 mi from the nearest point on a straight shore

nuais6lfp

nuais6lfp

Answered question

2021-11-19

A boat on the ocean is 4 mi from the nearest point on a straight shoreline; that point is 6 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. 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?
b. If she walks at 3 mi/hr, what is the minimum speed at which she must row so that the quickest way to the restaurant is to row directly (with no walking)?

Answer & Explanation

Unpled

Unpled

Beginner2021-11-20Added 23 answers

a) On the ocean, we have a boat that is 4mi on a straight stretch of shoreline, and we are aware that this point is 6mi distance from a waterfront restaurant. The activity begins with a  3mihr and rows at 2mihr and we must determine at which point on the shore she should land to minimize the total travel time. 
Let x represent the separation between the landing location and the closest coastline point. According to this, the distance between her landing spot and the restaurant is equivalent to (6 - x)mi.
Let t stand in for the amount of time it took her to go to the restaurant. It would be beneficial to represent t as a function of x (t = t(x)) and then equalize its derivative with zero in order to perform this minimization.
The distance between the boat and its landing is equal to thanks to the Pythagoras Theorem.
d=42+x2=16+x2
t(x)=16+x22+6x3hours 
For time to be minimal we place  dt  dx =0, which gives us: 
t(x)=2x416+x213=x216+x213=0 
3x=216+x2 and squaring both sides we get 
9x2=4(16+x2) or 5x2=64 
Since we are discussing distance, we must select the positive of the two solutions to this equation, which is
x=85mi=858mi
So, if she lands at the poin that is 6855mi=30855mi away from the restaurant, she will minimie the time needed to get to the restaurant.

memomzungup4

memomzungup4

Beginner2021-11-21Added 14 answers

b) We now know that she walks at 3mihr and we must determine the minimum speed at which she must row so that the quickest way to the restaurant is to row directly (without walking).
Since now we don't have the speed of rowing, the time will be:
t(x)=16+x2vr+6x3
Let's now determine vr from here and later on incorporate the condition that we have for x.
16+x2vr=t6x3=3tt+x3
vr=316+x23t6+x
For vr to be minimal we have that dvrdx=0.
dvrdx=3.
2x216+x2(3t6+x)16+x21(3t6+x)2=0, which implies that:
2x216+x2(3t6+x)16+x21=0
3tx6x+x216x216+x2=0, which imlies that
3tx6x+x216x2=0ort=16+6x3x (1)
Since we want to find the minimum speed of reaching the restaurant without walking, we have that x = 6, so if we incorporate that into equation (1) we get that t=16+3618=5218hr.
We now return vr and by incorporating t=5218hr we get that
vr=16+625218=525218=91313mihr
We conclude that the minimal speed at which she must row so that the quickest way to the restaurant is without walking is vr=91313mihr

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