Question

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.

Answer 1

Answer:

See explanation.

Step-by-step explanation:

The area $A_B$ of the bigger rectangle is its width $(x-2)$ times its height $2x+x$:

$A_B = (x-2)(2x+x)\:cm^2$

Similarly, the area $A_S$ of the smaller rectangle is

$A_S= x(x-4)\:cm^2.$

Together, we are told that the area of the compound shape is $36cm^2$; therefore,

$A_B+A_S= 36\:cm^2$

$(x-2)(2x+x)+ x(x-4)\:cm^2 = 36\:cm^2$

Expanding the left side of the equation, we get:

$4x^2 -10x =36\:$

$4x^2-10x -36 =0$.

Divide both side by 2, and we get:

$\boxed{ 2x^2-5x -18=0}$

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

$x = \dfrac{5\pm\sqrt{25-4(2)(-18)} }{4}$

$x = \dfrac{5\pm 13 }{4} \\\\x= -2\\\\x=4.5$

From these to solutions we pick the positive value $x =4.5$, since the lengths cannot be negative.

Since the length of AB is $x$, it is 4.5 cm:

$\boxed{AB =4.5 \: cm.}$