Math Help - angles problem

A man can row at 4 km/h, and run at 3 km/h. He needs to get from a point A, on the south bank of a stretch of still water, to point B on the north bank of the water. The direct distance from A to B is 5 km, and the water is 4 km wide. He starts rowing with an angle between North and the direction in which he rows. Find an expression for the time he will take to get from A to B, in terms of .

Hello, samtheman17!

A man can row at 4 km/h, and run at 3 km/h.
He needs to get from a point , on the south bank of a stretch of still water,
to point on the north bank of the water.
The direct distance from to is 5 km, and the water is 4 km wide.
He starts rowing with an angle θ between North and the direction in which he rows.

Find an expression for the time he will take to get from to , in terms of θ.

Code:

          3
    C * - - - o B
      |     *
    4 |   * 5
      | *
    A o

The width of the water is: km.
The distance from to is: km.
Pythagorus tells us that: km.


He will start at , row to , then run to

Code:

          x   P   3-x
    C * - - - * - - - - * B
      |      /
      |     /
      |    / y
    4 |   /
      |  /
      |θ/
      |/
    A *

Let
Let , then:
Let

In right triangle

He rows km at 4 km/hr.
. . This will take him: . hours.

Then he runs km at 3 km/hr.
.This will take him: . hours.

Hence, his total time is: . .[1]


We want this answer in terms of θ.

In right triangle .[2]

Note that: .

. . Hence: . .[3]


Subnstitute [2] and [3] into [1]: .


Therefore: .

thank you so much!

i was trying to do something like that but kept messing something up as there are multiple steps....

it is so clear now, thanks again