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 $\displaystyle 8\sqrt{2}$ *meters* northeast

of the tugboat and travels west at 7 cm/s.

How closely do the two tugboats approach each other?

Answer: 186 cm 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 $\displaystyle A$ and goes north at 5 cm/sec.

In $\displaystyle t$ seconds, it travels $\displaystyle 5t$ cm to $\displaystyle T.$

The launch starts at $\displaystyle C$ and goes west at 7 cm/sec.

In $\displaystyle t$ seconds, it travels $\displaystyle 7t$ cm to $\displaystyle L.$

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

. . $\displaystyle AB = BC = CD = DA = 800$

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

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

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

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

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

. . and that is the function we must minimize.

*Go for it!*

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

.