You live in a cabin exactly two miles north of a stream that flows east-west. Your grandmother lives 12 miles west and 1 mile north of your cabin. You want to visit your grandmother and take her some cold water from the stream. What is the shortest distance you must travel in order to leave your cabin, go to the stream, and take the water to your grandmother?
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to grandmothers house.
- Thread starter westworld
- Start date
You live in a cabin exactly two miles north of a stream that flows east-west. Your grandmother lives 12 miles west and 1 mile north of your cabin. You want to visit your grandmother and take her some cold water from the stream. What is the shortest distance you must travel in order to leave your cabin, go to the stream, and take the water to your grandmother?
Plot this on a coordinate system using the x-axis as the stream. Take shot at this and let us know if you get stuck.
Plotting on x/y axis I get 17 miles
How did you get 17 miles? That is not what I got. I got about 14.4 miles.
How did you get 17 miles? That is not what I got. I got about 14.4 miles.
Oh, I see just realized how you got 17. You went 2 miles to the stream from the house. Then 2 miles back to the house (4 total miles). Then 12 miles west (16 total miles) then 1 mile north (17 total miles). Your problem does not mention that you have to follow that path. It says we want the SHORTEST distance, so that would be 2 miles to the stream (perpindicular path), then at a diagonal to grandma's house which is the square root of 153 or about 12.4 miles (using good ol' Pythagorean theorem). Thus the total distance is about 14.4 miles in my humble opinion.
willmoore21
Junior Member
- Joined
- Jan 26, 2012
- Messages
- 75
Oh, I see just realized how you got 17. You went 2 miles to the stream from the house. Then 2 miles back to the house (4 total miles). Then 12 miles west (16 total miles) then 1 mile north (17 total miles). Your problem does not mention that you have to follow that path. It says we want the SHORTEST distance, so that would be 2 miles to the stream (perpindicular path), then at a diagonal to grandma's house which is the square root of 153 or about 12.4 miles (using good ol' Pythagorean theorem). Thus the total distance is about 14.4 miles in my humble opinion.
I did this and got the same, do you get it now?
Hello, westworld!
You need a better diagram . . .
The stream is \(\displaystyle DB\!:\;DB = 12\)
You are at \(\displaystyle A\!:\;AB = 2\)
Grandma is at \(\displaystyle C\!:\;CD = 1\)
We want to locate point \(\displaystyle P\)
. . so that the total distance \(\displaystyle AP + PC\) is a minimum.
Reflect point \(\displaystyle C\) across \(\displaystyle DB\) to point \(\displaystyle C'\)
Draw line \(\displaystyle AC'\)
It crosses \(\displaystyle DB\) at point \(\displaystyle P.\)
Point \(\displaystyle P\) is the desired point.
The straight line \(\displaystyle AC' \:=\:AP + PC'\) is the shortest distance from \(\displaystyle A\) to \(\displaystyle C'\)
From similiar right triangles, \(\displaystyle PC = PC'\)
Therefore: .\(\displaystyle AP + PC\) is the shortest distance.
In right triangle \(\displaystyle AEC'\!:\;(AC')^2 \:=\:12^2 + 3^2 \:=\:153\)
Therefore: .\(\displaystyle AP + PC \:=\:AC' \:=\:\sqrt{153} \:=\:3\sqrt{17} \:\approx\: 12.37 \text{ miles}\)
You need a better diagram . . .
You live in a cabin exactly two miles north of a stream that flows east-west.
Your grandmother lives 12 miles west and 1 mile north of your cabin.
You want to visit your grandmother and take her some cold water from the stream.
What is the shortest distance you must travel in order to leave your cabin,
go to the stream, and take the water to your grandmother?
Code:
* A
* |
C * * |
| * * | 2
1 | * * |
| * P * |
D *-------*-----------* B
: - - - 12 - - - - :You are at \(\displaystyle A\!:\;AB = 2\)
Grandma is at \(\displaystyle C\!:\;CD = 1\)
We want to locate point \(\displaystyle P\)
. . so that the total distance \(\displaystyle AP + PC\) is a minimum.
Reflect point \(\displaystyle C\) across \(\displaystyle DB\) to point \(\displaystyle C'\)
Draw line \(\displaystyle AC'\)
It crosses \(\displaystyle DB\) at point \(\displaystyle P.\)
Code:
* A
* |
C * * |
| * * | 2
1 | * * |
| * * |
D *-------*-----------* B
| * P :
1 | * : 1
| * :
C' * - - - - - - - - - * E
: - - - 12 - - - - :The straight line \(\displaystyle AC' \:=\:AP + PC'\) is the shortest distance from \(\displaystyle A\) to \(\displaystyle C'\)
From similiar right triangles, \(\displaystyle PC = PC'\)
Therefore: .\(\displaystyle AP + PC\) is the shortest distance.
In right triangle \(\displaystyle AEC'\!:\;(AC')^2 \:=\:12^2 + 3^2 \:=\:153\)
Therefore: .\(\displaystyle AP + PC \:=\:AC' \:=\:\sqrt{153} \:=\:3\sqrt{17} \:\approx\: 12.37 \text{ miles}\)
Hello, westworld!
You need a better diagram . . .
The stream is \(\displaystyle DB\!:\;DB = 12\)Code:* A * | C * * | | * * | 2 1 | * * | | * P * | D *-------*-----------* B : - - - 12 - - - - :
You are at \(\displaystyle A\!:\;AB = 2\)
Grandma is at \(\displaystyle C\!:\;CD = 1\)
We want to locate point \(\displaystyle P\)
. . so that the total distance \(\displaystyle AP + PC\) is a minimum.
Reflect point \(\displaystyle C\) across \(\displaystyle DB\) to point \(\displaystyle C'\)
Draw line \(\displaystyle AC'\)
It crosses \(\displaystyle DB\) at point \(\displaystyle P.\)
Point \(\displaystyle P\) is the desired point.Code:* A * | C * * | | * * | 2 1 | * * | | * * | D *-------*-----------* B | * P : 1 | * : 1 | * : C' * - - - - - - - - - * E : - - - 12 - - - - :
The straight line \(\displaystyle AC' \:=\:AP + PC'\) is the shortest distance from \(\displaystyle A\) to \(\displaystyle C'\)
From similiar right triangles, \(\displaystyle PC = PC'\)
Therefore: .\(\displaystyle AP + PC\) is the shortest distance.
In right triangle \(\displaystyle AEC'\!:\;(AC')^2 \:=\:12^2 + 3^2 \:=\:153\)
Therefore: .\(\displaystyle AP + PC \:=\:AC' \:=\:\sqrt{153} \:=\:3\sqrt{17} \:\approx\: 12.37 \text{ miles}\)
Soroban, the problem says that Grandma's house is 12 miles west and 1 mile north from THE CABIN. So the picture should have shown point D 12 miles directly west from point A and point C 1 mile north of point D. Then you must travel 2 miles from the house to the stream and then from there to Grandma's house on a straight line (hypotenuse of the triangle with sides 3 and 12). So the total distance (and thus the shortest distance) would be 2 + squareroot (153) or about 14.37 miles.
Or am I missing something here?