Note: Enter the answer and show all the steps that you use to solve this problem in the space provided.
A rectangle is shown with length x plus 10 and width 2 x plus 5. The inside of the rectangle is shaded other than an unshaded square with length x plus 1 and width x plus 1.
Write an expression for the area of the shaded region in its simplest form. Show all of your steps.
To find the area of the shaded region, we need to subtract the area of the unshaded square from the area of the rectangle.
The area of the rectangle is given by the formula: length * width.
Length of the rectangle = x + 10
Width of the rectangle = 2x + 5
So, the area of the rectangle is: (x + 10) * (2x + 5) = 2x^2 + 15x + 20x + 50 = 2x^2 + 35x + 50.
The area of the unshaded square is given by the formula: side * side.
The side of the unshaded square = x + 1.
So, the area of the unshaded square is: (x + 1) * (x + 1) = x^2 + 2x + x + 1 = x^2 + 3x + 1.
To find the area of the shaded region, we subtract the area of the unshaded square from the area of the rectangle.
Area of the shaded region = Area of the rectangle - Area of the unshaded square
= (2x^2 + 35x + 50) - (x^2 + 3x + 1)
= 2x^2 + 35x + 50 - x^2 - 3x - 1
= x^2 + 32x + 49.
Therefore, the expression for the area of the shaded region in its simplest form is x^2 + 32x + 49.
are you sure
Apologies for the confusion in my previous response. Let's go through the problem again to get the correct expression for the area of the shaded region.
The length of the rectangle is x + 10 and the width is 2x + 5.
The area of the rectangle is given by length * width, so the area of the rectangle is (x + 10) * (2x + 5).
The area of the unshaded square is given by (x + 1) * (x + 1) or (x + 1)^2.
To find the area of the shaded region, we need to subtract the area of the unshaded square from the area of the rectangle.
Area of the shaded region = Area of the rectangle - Area of the unshaded square
= (x + 10) * (2x + 5) - (x + 1)^2
= (2x^2 + 5x + 20x + 50) - (x^2 + 2x + 1)
= 2x^2 + 25x + 50 - x^2 - 2x - 1
= x^2 + 23x + 49.
Therefore, the correct expression for the area of the shaded region is x^2 + 23x + 49.
are you positive with this answer
Apologies for the confusion. Let's go through the problem once again to get the correct expression for the area of the shaded region.
The length of the rectangle is x + 10 and the width is 2x + 5.
The area of the rectangle is given by length * width, so the area of the rectangle is (x + 10) * (2x + 5).
The area of the unshaded square is given by (x + 1) * (x + 1) or (x + 1)^2.
To find the area of the shaded region, we need to subtract the area of the unshaded square from the area of the rectangle.
Area of the shaded region = Area of the rectangle - Area of the unshaded square
= (x + 10) * (2x + 5) - (x + 1)^2
Expanding the expressions, we get:
= (2x^2 + 5x + 20x + 50) - (x^2 + 2x + 1)
= 2x^2 + 25x + 50 - x^2 - 2x - 1
= x^2 + 23x + 49.
Therefore, the correct expression for the area of the shaded region is x^2 + 23x + 49.