Thread: 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 $\displaystyle A$, on the south bank of a stretch of still water,
to point $\displaystyle B$ on the north bank of the water.
The direct distance from $\displaystyle A$ to $\displaystyle B$ 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 $\displaystyle A$ to $\displaystyle B$, in terms of θ.

Code:

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

The width of the water is: $\displaystyle AC = 4$ km.
The distance from $\displaystyle A$ to $\displaystyle B$ is: $\displaystyle AB = 5$ km.
Pythagorus tells us that: $\displaystyle CB = 3$ km.


He will start at $\displaystyle A$, row to $\displaystyle P$, then run to $\displaystyle B.$

Code:

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

Let $\displaystyle \theta \,=\, \angle CAP$
Let $\displaystyle x \,=\, CP$, then: $\displaystyle 3-x \:=\:PB.$
Let $\displaystyle y \,=\,AP$

In right triangle $\displaystyle ACP\!:\;\;y^2 \:=\:x^2 + 4^2 \quad\Rightarrow\quad y \:=\:\sqrt{x^2+16}$

He rows $\displaystyle \sqrt{x^2+16}$ km at 4 km/hr.
. . This will take him: .$\displaystyle \frac{\sqrt{x^2+16}}{4}$ hours.

Then he runs $\displaystyle 3-x$ km at 3 km/hr.
.This will take him: .$\displaystyle \frac{3-x}{3}\:=\:1-\tfrac{1}{3}x$ hours.

Hence, his total time is: .$\displaystyle T \;=\;\tfrac{1}{4}\sqrt{x^2+16} + 1 -\tfrac{1}{3}x$ .[1]


We want this answer in terms of θ.

In right triangle $\displaystyle ACP\!:\;\tan\theta \:=\:\frac{x}{4} \quad\Rightarrow\quad x \:=\:4\tan\theta$ .[2]

Note that: .$\displaystyle x^2+16 \;=\;(4\tan\theta)^2 + 16 \;=\;16\tan^2\!\theta + 16 \;=\;16(\tan^2\!\theta+1) \;=\;16\sec^2\!\theta $

. . Hence: .$\displaystyle \sqrt{x^2+16} \;=\;4\sec\theta$ .[3]


Subnstitute [2] and [3] into [1]: .$\displaystyle T \;=\;\tfrac{1}{4}(4\sec\theta) + 1 - \tfrac{1}{3}(4\tan\theta)$


Therefore: .$\displaystyle T \;=\;\sec\theta + 1 - \tfrac{4}{3}\tan\theta$

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