Question

An island is 3 kilometers due north of its closest point along a straight shoreline. A visitor is staying in a tent that is 13 kilometers east of that point. The visitor is planning to go from the tent to the island. Suppose the visitor runs at a rate of 8 kmph and swims at a rate of 1 km. How far should the visitor run to minimize the time it takes to reach the island

Answer 1

Run to the tent and then swim to island

Explanation:

Given:-

- Position of island, N = 3 km North of point on shoreline

- Position of tent, E = 13 km East of point on shoreline

- Running speed, r = 8 km/h

- Swimming speed, s = 1 km/h

Find:-

How far should the visitor run to minimize the time it takes to reach the island

Solution:-

- The shortest distance i.e (displacement) L of island from tent can be determined by pythagorean theorem:

`L^2 = P^2 + B^2\\\\L = √(P^2 + B^2)\\\\L = √(13^2 + 3^2) = 13.34166 km`

- The time taken (t1) for the person to swim directly from tent to island over the distance (L) is given by:

t1 = L / s

t1 = 13.34166 / 1 = 13.3 hrs

- However if the visitor was to first runs to the closest point on the shoreline to the island, then swims across to the island, it will take time (t2);

t2 = E/r + N/s

t2 = 13/8 + 3/1

t2 = 4.625 hr.

- Therefore, to minimize the time it takes to reach the island, the visitor has to run 13 km west of the tent to first get to the closest point of the shoreline to the island before swimming across to the island.

Answer 2

The visitor should run 13 km due west from the tent

Explanation:

Here we have;

Location of island = 3 km North of point on shoreline

Location of tent = 13 km East of point on shoreline

Running speed of visitor = 8 km/h

Swimming speed = 1 km/h

Distance of island from tent =
`\sqrt{3^(2) + 13^(2) } = 13.34 km`

Since, time = distance/speed, it will take 13.34/1 hours or 13.34 hours to swim directly to the island.

However if the visitor first runs to the closest point on the shoreline to the island, then swims across to the island, it will take;

13/8 Hr + 3/1 hr = 37/8 hours or 4.625 hours only.

Therefore, to minimize the time it takes to reach the island, the visitor has to run 13 km west of the tent to first get to the closest point of the shoreline to the island before swimming across to the island.