# Thread: Optimization Problems (time & distance)

1. ## Optimization Problems (time & distance)

Hi everyone, I'm having some trouble with these 2 problems. Hope you can help me out.

1. A toy tugboat is launched from the side of a pond and travels north at 5 cm/s. At the same moment, a toy launch starts from a point 8root2 m northeast of the tugboat and travels west at 7 cm/s. How closely to the two tugboats approach eachother.

2. Two isolated farms are situated 12 km apart on a straight country road that runs parallel to the main highway 20 km away. The power company decides to run a wire from the highway to a junctionbox, and from there, wires of equal length to the two houses. Where should the junction box be placed to minimize the length of wires needed?

I can't seem to think of these questions the right way. Any help would be very appriciated.

Thanks

2. Originally Posted by hsidhu
...
2. Two isolated farms are situated 12 km apart on a straight country road that runs parallel to the main highway 20 km away. The power company decides to run a wire from the highway to a junctionbox, and from there, wires of equal length to the two houses. Where should the junction box be placed to minimize the length of wires needed?

...
1. Draw a rough sketch of the situation (see attachment)

2. The total length of the used wire is:

$l = x + 2 w$

3. Since $w^2 = 6^2+(20-x)^2 = 436-40x+x^2$ you'l get:

4. $l(x)=x+2\cdot \sqrt{436-40x+x^2}$

5. Solve the equation $l'(x) = 0$ for x. (Don't forget to use the chainrule when deriving the function l)

6. I've got $x = 20 - 2·\sqrt{3} \approx 16.5359$

3. Originally Posted by hsidhu
Hi everyone, I'm having some trouble with these 2 problems. Hope you can help me out.

1. A toy tugboat is launched from the side of a pond and travels north at 5 cm/s. At the same moment, a toy launch starts from a point 8root2 m northeast of the tugboat and travels west at 7 cm/s. How closely to the two tugboats approach eachother.

...
1. Draw a sketch (see attachment). The boats are sailing on the legs of an isosceles right triangle whose legs are 8 m long.

2. Let t denote the elapsed time in seconds and transform the length of 8 m into 800 cm. The distance d is the hypotenuse of a right triangle with the legs (800 - 5t) and (800 - 7t) respectively.

3. Thus:

$(d(t))^2=(800-5t)^2+(800-7t)^2 = 74·t^2 - 19200·t + 1280000$

4. If d(t) has an extreme value then the function $D(t)=(d(t))^2$ has a maximum. Consider the function D. Calculate the first derivation and solve D(t)' = 0 for t.

5. Plug in this value into $d(t)=\sqrt{74·t^2 - 19200·t + 1280000}$.

I've got $d\left( \dfrac{4800}{37} \right) = \dfrac{800·\sqrt{74}}{37} \approx 185.9962219$

4. Hello, hsidhu!

The first one is tricky to set up . . .

1. A toy tugboat is launched from the side of a pond and travels north at 5 cm/s.
At the same moment, a toy launch starts from a point $8\sqrt{2}$ meters northeast
of the tugboat and travels west at 7 cm/s.
How closely do the two tugboats approach each other?

Code:
800-7t L       7t
B o - - - o - - - - - - - o C
|      /              * |
|     /             *   |
|    / d          *     |
800-5t|   /       _   *       |
|  /    800√2 *         |
| /         *           | 800
|/        *             |
T o       *               |
|     *                 |
5t |   *                   |
| * 45°                 |
A o - - - - - - - - - - - o D
800

The tugboat starts at $A$ and goes north at 5 cm/sec.
In $t$ seconds, it travels $5t$ cm to $T.$

The launch starts at $C$ and goes west at 7 cm/sec.
In $t$ seconds, it travels $7t$ cm to $L.$

Since $AC = 800\sqrt{2}$, it is the diagonal of an 800-cm square $ABCD.$
. . $AB = BC = CD = DA = 800$

Hence: . $BT \:=\: 800-5t,\;\;BL \:=\: 800-7t$

We want to minimize distance: . $d \:=\:TL$

In right triangle $LBT\!:\;\;d \:=\:\sqrt{(800-5t)^2 + (800-7t)^2}$

Let $D \:=\:d^2 \:=\:(800-5t)^2 + (800-7t)^2$

We have: . $D \;=\;1,\!280,\!000 - 19,\!200t + 74t^2$

. . and that is the function we must minimize.

Go for it!

Edit: E.B. beat me to it . . . *sigh*
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