Question

Rectangle A has a length of 2x + 6 and a width of 3x. Rectangle B has a length of x + 2 and an area of 12 square units greater than Rectangle A’s area. What is a simplified expression for the width of Rectangle B?

x + 2

x + 1

6x + 6

6(x + 2)(x + 1)

Answer 1

are you sure

Answer 2

wrong

Answer 3

try using the given choicees and see which fits the quastion

Answer 4

try using the given choices and plug into rectangle B width and solve again

Answer 5

x + 1

Answer 6

I apologize for the error in my previous response. Let's correct it:

The area of Rectangle A is given by length*width: (2x + 6)(3x) = 6x^2 + 18x.

The area of Rectangle B is 12 square units greater than Rectangle A's area, so Rectangle B's area is: 6x^2 + 18x + 12.

The length of Rectangle B is x + 2, and the width of Rectangle B is the unknown we are trying to find, let's call it w.

So the area of Rectangle B should be length*width: (x + 2)(w).

We can set up the equation:
(x + 2)w = 6x^2 + 18x + 12

We know that w = 3x because that is the width of Rectangle A.

Therefore: (x + 2)(3x) = 6x^2 + 18x + 12
3x^2 + 6x = 6x^2 + 18x + 12
0 = 3x^2 + 12x + 12
0 = 3(x^2 + 4x + 4)

Factoring, we get:
0 = 3(x + 2)^2

Thus, the simplified expression for the width of Rectangle B is 3(x + 2), which simplifies to 3x + 6.

Therefore, the correct simplified expression for the width of Rectangle B is:
3x + 6.

Answer 7

I apologize for the mistake. Let's correct the solution.

Given:
Rectangle A:
Length = 2x + 6
Width = 3x

Rectangle B:
Length = x + 2
Area = Area of Rectangle A + 12
Width = unknown (w)

Area of Rectangle A = (2x + 6)(3x) = 6x^2 + 18x
Area of Rectangle B = Area of Rectangle A + 12 = 6x^2 + 18x + 12

The area of Rectangle B is given by Length*Width: (x + 2)(w) = 6x^2 + 18x + 12

(x + 2)w = 6x^2 + 18x + 12

Let's substitute w = 3x (Width of Rectangle A) into the equation:
(x + 2)(3x) = 6x^2 + 18x + 12
3x^2 + 6x = 6x^2 + 18x + 12
0 = 3x^2 + 12x + 12
0 = 3(x^2 + 4x + 4)
0 = 3(x + 2)(x + 2)

Therefore, the simplified expression for the width of Rectangle B is:
3(x + 2)

Hence, the correct simplified expression for the width of Rectangle B is 3(x + 2), which simplifies to 3x + 6. Thank you for your understanding.

Answer 8

Let's try using the given choices:

Given choices:
- x + 2
- x + 1
- 6x + 6
- 6(x + 2)(x + 1)

We know that Rectangle A's width is 3x.

We need the width of Rectangle B, which is unknown and denoted as w.

Rectangle B's area is 12 square units greater than Rectangle A's area, so:
Area of Rectangle A: 6x^2 + 18x
Area of Rectangle B: 6x^2 + 18x + 12

Let's find w by setting up the equation:
(x + 2)w = 6x^2 + 18x + 12

Now, we substitute Rectangle A's width, which is 3x:
(x + 2)(3x) = 6x^2 + 18x + 12
3x^2 + 6x = 6x^2 + 18x + 12
-3x^2 - 12x - 12 = 0
-3(x^2 + 4x + 4) = 0

Factoring, we get:
-3(x + 2)(x + 2) = 0

The simplified expression for the width of Rectangle B is 3(x + 2), which equals 3x + 6 after simplifying.

None of the given choices match the simplified expression for the width of Rectangle B, which is 3x + 6, after solving the problem. Thank you for pointing that out.

Answer 9

Let's do as you suggested:

Given choices:
- x + 2
- x + 1
- 6x + 6
- 6(x + 2)(x + 1)

Rectangle A:
Length = 2x + 6
Width = 3x
Area = (2x + 6)(3x) = 6x^2 + 18x

Rectangle B:
Length = x + 2
Width = w (to be found)
Area of Rectangle B = Area of Rectangle A + 12
(x + 2)w = 6x^2 + 18x + 12

Let's test each choice for the width of Rectangle B:

1. Using x + 2:
(x + 2) * (x + 2) = x^2 + 4x + 4
This does not match the Area of Rectangle B, which is 6x^2 + 18x + 12.

2. Using x + 1:
(x + 2) * (x + 1) = x^2 + 3x + 2
This also does not match the Area of Rectangle B.

3. Using 6x + 6:
(x + 2) * (6x + 6) = 6x^2 + 12x + 12
This also does not match the Area of Rectangle B.

4. Using 6(x + 2)(x + 1):
6(x + 2)(x + 1) = 6x^2 + 18x + 12
This matches the Area of Rectangle B.

Therefore, the correct choice for the width of Rectangle B from the given options is:
6(x + 2)(x + 1)