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The diagram shows two joined rectangles. The total area of the compound shape ABCDEF is 36 cm^2. By considering the areas of the two rectangles, show that
2x^2 - 5x - 18 = 0 and hence find the value of length AB.

The diagram shows two joined rectangles The total area of the compound shape ABCDEF is 36 cm2 By considering the areas of the two rectangles show that 2x2 5x 18 class=

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Poltergeist

Answer:

See explanation.

Step-by-step explanation:

The area [tex]A_B[/tex] of the bigger rectangle is its width [tex](x-2)[/tex] times its height [tex]2x+x[/tex]:

[tex]A_B = (x-2)(2x+x)\:cm^2[/tex]

Similarly, the area [tex]A_S[/tex] of the smaller rectangle is

[tex]A_S= x(x-4)\:cm^2.[/tex]

Together, we are told that the area of the compound shape is [tex]36cm^2[/tex]; therefore,

[tex]A_B+A_S= 36\:cm^2[/tex]

[tex](x-2)(2x+x)+ x(x-4)\:cm^2 = 36\:cm^2[/tex]

Expanding the left side of the equation, we get:

[tex]4x^2 -10x =36\:[/tex]

[tex]4x^2-10x -36 =0[/tex].

Divide both side by 2, and we get:

[tex]\boxed{ 2x^2-5x -18=0}[/tex]

Using the quadratic equation the solution to this equation we get are:

[tex]x = \dfrac{5\pm\sqrt{25-4(2)(-18)} }{4}[/tex]

[tex]x = \dfrac{5\pm 13 }{4} \\\\x= -2\\\\x=4.5[/tex]

From these to solutions we pick the positive value [tex]x =4.5[/tex], since the lengths cannot be negative.

Since the length of AB is [tex]x[/tex], it is 4.5 cm:

[tex]\boxed{AB =4.5 \: cm.}[/tex]

aksnkj

The length of side AB is x or [tex]\dfrac{9}{2}[/tex] cm.

Given information:

The area of the shape ABCDEF is 36 square centimeter.

From the given figure:

  • In the smaller rectangle, length of sides AB is x and BC is (2-4) cm.
  • In the larger rectangle, length of the sides DE is (x-2) and EF is 3x cm.

Now, the area of the smaller rectangle will be,

[tex]A_s=x(x-4)\\=x^2-4x[/tex]

The area of the larger rectangle will be,

[tex]A_l=3x(x-2)\\=3x^2-6x[/tex]

The total area of the shape can be written as,

[tex]A_s+A_l=36\\x^2-4x+3x^2-6x=36\\4x^2-10x-36=0\\2x^2-5x-18=0[/tex]

So, [tex]2x^2 - 5x - 18 = 0[/tex] quadratic equation is correct.

Now, solve the quadratic equation for x as,

[tex]2x^2 - 5x - 18 = 0\\2x^2 - 9x+4x - 18 = 0\\x(2x-9)+2(2x-9)=0\\(2x-9)(x+2)=0\\x=-2,\dfrac{9}{2}[/tex]

The length of the side cannot be negative.

Therefore, the length of side AB is x or [tex]\dfrac{9}{2}[/tex] cm.

For more details refer to the link:

https://brainly.com/question/21982865

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