An island is 3 mi due north of its closest point along a straight shoreline. you are staying at a cabin on the shore that is 7 mi west of that point. you are planning to go from the cabin to the island. suppose you run at a rate of 9 mph and row at a rate of 5 mph. how far should you run (in miles) before rowing to minimize the time it takes to reach the island

  5.00 miles

Step-by-step explanation:

Let x represent the distance in miles you should run. Then the distance you will be rowing is ...

  √(3² +(7-x)²)

and your total travel time in hours is ...

  t = x/9 + √(3² +(7-x)²)/5

This is minimized when its derivative with respect to x is zero.

This will be zero when the numerator is zero, so ...

Solving this quadratic by your favorite method gives ...

  x ≈ 4.996 . . . . . . there is an extraneous solution at x ≈ 9

You should run 5.00 miles before rowing in order to minimize the time to reach the island.

_____

Generic solution

For travel speeds a and b, where a < b and b represents the speed along the shore, the distance from the point nearest the island is given by tan(arcsin(a/b)) times the distance to the island.

Here, that is (3 mi)(tan(arcsin(5/9)) ≈ 2.004 miles. Since we're running from a point 7 miles from the point nearest the island, our running distance is 7 -2.004 = 4.996 miles.

If the starting point is less than the distance computed above, then the shortest time path is a straight line to the island.

In short, the travel angle from a line perpendicular to shore is given by arcsin(a/b).

This same solution works for problems involving laying pipeline, walking through woods, or any other scenario where there is an optimal straight-line path to a point where the cost changes.

.24

step-by-step explanation:

the probability is   .24

the first player had the highest probability of getting a silver marble

the game could be made fair by adding back the marble that was drawn after each draw

b) 1/3 * 1/5

step-by-step explanation:

each smallest part is 1/15 since the written one is 1/5 and it's divided by 3 equal parts. (1/5 )/3 = 1/15 = 1/3 * 1/5