Answer:
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
[tex]y^2-x^3+x-3=0[/tex]
We can convert this to a explicit form
[tex]y=\sqrt{x^3-x+3}[/tex]
The function D (distance) we have to minimize can be expressed as
[tex]D=\sqrt{(y-y_0)^2+(x-x_0)^2}\\\\D=\sqrt{(\sqrt{x^3-x+3}-1)^2+(x-4)^2}[/tex]
As the distance is always positive, for simplicity we can derive D² and still get the same result.
[tex]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}[/tex]
The point that minimizes the distance is the one that satisfies
[tex]d(D^2)/dx=0[/tex]
[tex]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[/tex]
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.
[tex]y=\sqrt{x^3-x+3}=\sqrt{1.702^3-1.702+3}=\sqrt{4.930-1.702+3}=2.496[/tex]
In the graph you can see
- The camp (in green)
- The river (in red)
- The derivative of the square of the distance (in black)
The shortest distance from the camp to the river is given by the minimum distance between two points
Reason:
The coordinate of the location of Everett, Delmar, and Pete = (4, 1)
The equation that describes the curve of the river is y² - x³ + x - 3 = 0
The point of the river that they should get their water in order to minimize their distance travelled
The distance of a point from a curve
The point D of minimum distance is given as follows;
√((x - 4)² + (√(x³ - x + 3) - 1)²)
Squaring both sides gives;
Differentiating gives;
[tex]\dfrac{d \left(D^2 \right)}{dx} = \dfrac{d}{dx} \left((x - 4)^2 + (\sqrt{x^3 - x + 3} - 1)^2 \right) = 3\cdot x^2 + 2 \cdot x - 9 - 2 \cdot \dfrac{3 \cdot x^2 - 1}{2 \cdot \sqrt{x^3-x + 3} }[/tex]
[tex]\dfrac{d \left(D^2 \right)}{dx} = 0 = 3\cdot x^2 + 2 \cdot x - 9 - 2 \cdot \dfrac{3 \cdot x^2 - 1}{2 \cdot \sqrt{x^3-x + 3} }[/tex]
Therefore, we have;
Solving with a graphing calculator gives the solution as x ≈ 1.70071
Which gives;
Therefore;
Learn more about shortest distance from a point to a curve here:
https://brainly.com/question/2264602
