A boat on the ocean is 5 mi from the nearest point on a straight shoreline; that

Question

A boat on the ocean is 5 mi from the nearest point on a straight shoreline; that point is 15 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 rai/hr and rows at 2 mi/hr, at which point on the shore should she land to minimize the total travel time? The boat should land miles from the restaurant. (Type an exact answer, using radicals as needed.) b. If she walks at 3 milk, 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)? The minimum speed she must row is n mi1hr.

Explanation / Answer

1. first lets write the time equation.

let the distance from hotel to the point where she should land for the minimal time be x

we know that time taken= distance /speed

so T = (sqrt(52+(15-x)2)/2) + x/3

(sqrt(52+(15-x)2)/2) this is the time taken for rowing and x/3 is time taken for walking.

time should be minimum, so dt/dx should be equal to 0

so dt/dx= 1/3 - ((15-x)/(2.sqrt((15-x)2+25)))

when dt/dx= 0

we get on solving,

5(15-x)2-100 = 0

we get roots

15+sqrt(20) and 15-sqrt(20)

so for minimum time the boat should land 15-sqrt(20) miles from the restaurant.

15-4sqrt(5) miles.

2. if she walks at 3mi/hr we should calculate the time taken for the direct travel thatid the hypotenuse in the values of speed.

take speed as x and the equation for t should be minimum so

dt/dx = 0

and we get speed.

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