writing expressions

Question

Question - Writing an expression that representsd the area of the shaded region, and evaluate that expression for the valuesof X in the table:

 

 

Its a rectangle

 

outside the rectangle  Length = 2x +5

                                Width = 4x +2

 

Inside the rectangle there is a square with dimensions of length = x+1

                                                                                  width = x+2

 

The other area is shaded all around it.

 

 

The table is

 

X              Area

3

5

10

20

 

Answer 1

To solve this, you must first create the expression that represents the area of the two shapes.  

The original rectangle has the side lengths of 2x+5 and 4x+2.  Area is length x width.  

Area= (2x + 5)(4x + 2) or 2x(4x +2) + 5(4x +2) -Use the distributive property
Area= 8x^2 + 4x + 20x + 10 - combine like terms
Area= 8x^2 + 24x + 10

You will need to do the same for the square.

 

The original square has the side lengths of x+1 and x+2. Area is length x width.

 

Area= (x+1)(x+2) or x(x+2) + 1(x+2) - use the distributive property
Area= x^2+ 2x + 1x + 2 - combine like terms
Area= x^2 + 3x + 2

 

Now you must subtract the area of the square (which is unshaded) from the area of the whole rectangle.  

 

Area= (8x^2 + 24x + 10) - (x^2 + 3x +2) - distribute the negative to all numbers in the second parenthesis. 

Area= 8x^2 + 24x + 10 + (-x^2) + (-3x) + (-2)- now combine like terms
Area= 7x^2 + 21x +8

 

Use this final equation to find the area for the shaded area and complete the table.

 

For example, if x=3, plug 3 in for all x's within the equation and follow the order of operations (PEMDAS)

 

Area= 7(3)^2 + 21(3) +8
Area= 7(9) + 21(3) + 8
Area= 63 + 63 + 8
Area= 126 +8

Area= 134