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.
Answer: 186 cm
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?
Answer: 16.5 km from the main road
I can't seem to think of these questions the right way. Any help would be very appriciated.
Thanks
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:
4. If d(t) has an extreme value then the function has a maximum. Consider the function D. Calculate the first derivation and solve D(t)' = 0 for t.
5. Plug in this value into .
I've got
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 meters northeast
of the tugboat and travels west at 7 cm/s.
How closely do the two tugboats approach each other?
Answer: 186 cmCode: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 and goes north at 5 cm/sec.
In seconds, it travels cm to
The launch starts at and goes west at 7 cm/sec.
In seconds, it travels cm to
Since , it is the diagonal of an 800-cm square
. .
Hence: .
We want to minimize distance: .
In right triangle
Let
We have: .
. . and that is the function we must minimize.
Go for it!
Edit: E.B. beat me to it . . . *sigh*
.