Question

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 (see figure). a woman plans to row the boat straight to a point on the shore and then walk along the shore to the restaurant. 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? 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 1

To minimize the total travel time, the woman should land at the point on the shore where the perpendicular distance from the boat to the restaurant is minimized.

Step-by-step explanation:

To minimize the total travel time, the woman should land at the point on the shore where the perpendicular distance from the boat to the restaurant is minimized. This point can be determined using the concept of similar triangles.

Let x be the distance from the boat to the desired landing point on the shore. Using similar triangles, we can set up the following proportion: (4 - x) / 6 = x / (6 / 3).

By solving this proportion, we can find the value of x.

To find the minimum speed at which she must row to have the quickest way to the restaurant by rowing directly, we can set up the equation: time rowing = distance / rowing speed.

The rowing distance can be calculated using the Pythagorean theorem: √((4 - x)^2 + 6^2).

By substituting the rowing distance and rowing speed into the equation, we can solve for time rowing. We can then compare this time to the time it takes to walk along the shore to determine the quicker option.