Question

FIND THE AREA OF THE SHADED REGION. SHOW ALL YOUR WORK.

Answer 1

The teachers this is commonly known as a doughnut problem. Basically you find the area of the large trays and subtract the area of the small shape.

Lg. Rectangle. Sm. Rectangle
A = lw. A=lw
A=(3x - 4)(2x + 2) A=(x - 3)(x - 6). Use FOIL on each
A = 6x^2 + 6x - 8x -8 A=x^2 - 6x - 3x + 18. Combine like terms
A = 6x^2 - 2x - 8. A=x^2 - 9x + 18. Subtract rectangles to find area of shaded region
A = 6x^2 -2x - 8 - (x^2 - 9x + 18) Distribute the negative into the parenthesis (multiply each term by -1)
A = 6x^2 - 2x - 8 - x^2 + 9x -18. Combine like terms
A = 5x^2 + 7x - 26. <— answer

Answer 2

The area of the shaded region is $5x^2 + 7x -26$

The dimension of the big rectangle is 3x - 4 by 2x + 2.

So, the area of the big rectangle is:

$A = (3x - 4) * (2x + 2)$

Multiply

$A = 6x^2 + 6x - 8x - 8$

$A = 6x^2 - 2x - 8$

The dimension of the small rectangle is x - 3 by x - 6

So, the area of the small rectangle is:

$A = (x - 3)(x - 6)$

Multiply

$A = x^2 - 3x - 6x + 18$

$A = x^2 -9x + 18$

The area of the shaded region is the difference between the above areas.

So, we have:

$A = 6x^2 - 2x - 8 - x^2 + 9x -18$

Collect like terms

$A = 6x^2 - x^2 - 2x + 9x -18- 8$

$A = 5x^2 + 7x -26$

Hence, the area of the shaded region is $5x^2 + 7x -26$

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