Mathematics, 02.11.2019 03:31 fabiolabaritto

Everett, delmar, and pete15 are at their camp at the point (4,1). they need to fetch water from theriver whose curve is described by the equation y2−x3+x−3 = 0 and then return to camp. at what point on the river should they get their water in order to minimize their distance travelled? you may need to use matlab and or mathematic

Answers

Answer from: madinaxoxo605

The point that minimize the distance is (x=1.702;y=2.496)

Step-by-step explanation:

In this problem we need to minimize the distance from a point (the camp) to a curve (the river).

The river follows the implicit function

y^2-x^3+x-3=0

We can convert this to a explicit form

$y=\sqrt{x^3-x+3}$

The function D (distance) we have to minimize can be expressed as

$D=\sqrt{(y-y_0)^2+(x-x_0)^2}\\\\D=\sqrt{(\sqrt{x^3-x+3}-1)^2+(x-4)^2}$

As the distance is always positive, for simplicity we can derive D² and still get the same result.

$D^2=(\sqrt{x^3-x+3}-1)^2+(x-4)^2\\\\D^2=(x^3-x+3)-2\sqrt{x^3-x+3}+1+x^2-8x+16\\\\D^2=x^3+x^2-9x+17-2\sqrt{x^3-x+3}$

The point that minimizes the distance is the one that satisfies

d(D^2)/dx=0

$d(D^2)/dx=d(x^3+x^2-9x+17)/dx-d(2\sqrt{x^3-x+3})/dx=0\\\\3x^2+2x-9-2(\frac{1}{2} \frac{1}{\sqrt{x^3-x+3}} *(3x^2-1))=0\\\\3x^2+2x-9- \frac{3x^2-1}{\sqrt{x^3-x+3}} =0$

This equation has a solution in x=1.702 (solved graphically).

This corresponds to the point (x=1.702;y=2.496) of the river.

$y=\sqrt{x^3-x+3}=\sqrt{1.702^3-1.702+3}=\sqrt{4.930-1.702+3}=2.496$

In the graph you can see

- The camp (in green)

- The river (in red)

- The derivative of the square of the distance (in black)

Everett, delmar, and pete15 are at their camp at the point (4,1). they need to fetch water from ther

Answer from: Quest

The answer is d 18ft

Answer from: Quest

Ibelieve it's a. 2.2 - x = -10.85 (plz tell me if it's right : ) )

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Everett, delmar, and pete15 are at their camp at the point (4,1). they need to fetch water from ther...