Respuesta :
Okay, so first you draw a picture and let x be the distance from point D to the rest stop. Then the distance from point to the rest stop is 8 - x
You know that the length of the new trail is y + z, where y is the distance from Ancaster to the rest stop and z is the distance from Dundas to the rest stop.
Now by the Pythagorean theorem, y^2 = 4^2 + x^2 and z^2 = 6^2 + (8 - x) ^2
So take square roots of these, add them, and minimize.
Note: I am assuming the path is perfectly straight, otherwise this approach fails.
You know that the length of the new trail is y + z, where y is the distance from Ancaster to the rest stop and z is the distance from Dundas to the rest stop.
Now by the Pythagorean theorem, y^2 = 4^2 + x^2 and z^2 = 6^2 + (8 - x) ^2
So take square roots of these, add them, and minimize.
Note: I am assuming the path is perfectly straight, otherwise this approach fails.
The rest stop should be built to minimize the length of a new trail that must be built from Ancaster and Dundas towns to the rest stop at 3.2 km.
What is Pythagoras theorem?
Pythagoras theorem says that in a right angle triangle the square of hypotenuse side is equal to the sum of square of other two legs of right angle triangle.
Let the distance from rest stop to point C is x and from Ancaster town to rest stop is [tex]d_1[/tex]. Thus, by the pythagours therorem
[tex]d_1=\sqrt{x^2+4^2}\\d_1=\sqrt{x^2+16}[/tex]
Similarly, let from Dundas town to rest stop is [tex]d_2[/tex]. The distance from rest stop to point D is (x-8). Thus, by the pythagours therorem
[tex]d_2=\sqrt{(8-x)^2+6^2}\\d_2=\sqrt{64+x^2-16x+36}\\d_2=\sqrt{x^2-16x+100}[/tex]
Total distnace,
[tex]d=\sqrt{x^2+4}+\sqrt{x^2-16x+100}[/tex]
Differenciate the above equation with respect to x, we get,
[tex]\dfrac{d(d)}{dx}=\dfrac{x}{\sqrt{x^2+16}}+\dfrac{x-8}{\sqrt{x^2-16x+100}}[/tex]
Equating the euqation equal to zero and solveing further we get,
[tex]\dfrac{x}{\sqrt{x^2+16}}+\dfrac{x-8}{\sqrt{x^2-16x+100}}=0\\\dfrac{x}{\sqrt{x^2+16}}=\dfrac{x-8}{\sqrt{x^2-16x+100}}\\20x^2+256x-1024=0[/tex]
Solving this quadratic equation, we get the values of x as 16/5 and 16. In this values 16/5 is shortest which is equal to 3.2.
Hence, the rest stop should be built to minimize the length of a new trail that must be built from Ancaster and Dundas towns to the rest stop at 3.2 km.
Learn more about the Pythagoras theorem here;
https://brainly.com/question/343682
