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Asked By :  Alejandro
Answers1

A point on yx2y sqrtx2 on quadrant 2 alpha a point on y2xy

A point on y=x+2y = \sqrt{x+2} on quadrant 2 (α\alpha)
A point on y=2xy = \sqrt{2-x} on quadrant 1 (β\beta)
And 2 other points on the x-axis (γ,δ\gamma, \delta)

Max length of the perimeter of the rectangle with α,β,γ,δ\alpha, \beta, \gamma, \delta as its vertices = LL

Rectangle area = SS

Find 8SL=?\frac{8S}{L} = ?

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Answers :

0

Let’s solve the given problem step-by-step.

Problem Overview

We aim to find the value of 8SL\frac{8S}{L}, where:

  • SS is the area of the rectangle.
  • LL is the maximum perimeter of the rectangle formed by vertices α\alpha, β\beta, γ\gamma, and δ\delta.

The points α\alpha and β\beta lie on the curves:

  1. y=x+2y = \sqrt{x+2} (Quadrant II).
  2. y=2xy = \sqrt{2-x} (Quadrant I).

The points γ\gamma and δ\delta lie on the x-axis.

Step 1: Coordinates of the Points

  1. α=(a,a+2)\alpha = (-a, \sqrt{-a + 2}) for a2a \leq -2.
  2. β=(b,2b)\beta = (b, \sqrt{2 - b}) for b2b \leq 2.

Step 2: Dimensions of the Rectangle

The height of the rectangle is determined by the yy-coordinates of α\alpha and β\beta: h=a+2=2b.h = \sqrt{-a + 2} = \sqrt{2 - b}.

The width of the rectangle is determined by the distance between γ\gamma and δ\delta, which is: w=ba.w = b - a.

Step 3: Area of the Rectangle

The area of the rectangle SS is given by:

S=wh=(ba)a+2.S = w \cdot h = (b - a) \cdot \sqrt{-a + 2}.

Step 4: Perimeter of the Rectangle

The perimeter LL of the rectangle is:

L=2(w+h)=2[(ba)+a+2].L = 2(w + h) = 2[(b - a) + \sqrt{-a + 2}].

Step 5: Maximize the Perimeter LL

We maximize LL by choosing values for aa and bb such that the rectangle covers the largest possible area under the curves. In this scenario:

  • The maximum width ww is achieved when a=2a = -2 and b=2b = 2.

Thus:

w=ba=2(2)=4.w = b - a = 2 - (-2) = 4.

The corresponding height hh is:

h=a+2=4=2.h = \sqrt{-a + 2} = \sqrt{4} = 2.

Step 6: Compute SS and LL

Using the maximum dimensions:

  1. Area:

S=wh=42=8.S = w \cdot h = 4 \cdot 2 = 8.

  1. Perimeter:

L=2(w+h)=2(4+2)=12.L = 2(w + h) = 2(4 + 2) = 12.

Step 7: Compute 8SL\frac{8S}{L}

8SL=8812=6412=163.\frac{8S}{L} = \frac{8 \cdot 8}{12} = \frac{64}{12} = \frac{16}{3}.

Final Answer:

163\boxed{\frac{16}{3}}


Answered By

Diego

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