One rectangle is twice as long as it is wide. The second rectangle is 3cm longer than the first one, and 1cm narrower than the first one. The areas of both rectangles are the same. What are the dimensions of the first rectangle?
L1 = Length of the first rectangle
W1 = Width of the first rectangle
A1 = Area of the first rectangle
L2 = Length of the second rectangle
W2 = Width of the second rectangle
A1 = Area of the second rectangle
L1 = 2 W1
A1 = L1 * W1 = 2 W1 * W1 = 2 W1 ^ 2
L2 = L1 + 3 = 2 W1 + 3
W2 = W1 - 1
A2 = L2 * W2 = ( 2 W1 + 3 ) * ( W1 - 1 )
A1 = A2
2 W1 ^ 2 = ( 2 W1 + 3 ) * ( W1 - 1 )
2 W1 ^ 2 = 2 W1 * W1 + 3 * W1 - 1 * 2 W1 - 1 * 3
2 W1 ^ 2 = 2 W1 ^ 2 + 3 W1 - 2 W1 - 3
2 W1 ^ 2 = 2 W1 ^ 2 + W1 - 3 Subtract 2 W1 ^ 2 to both sides
2 W1 ^ 2 - 2 W1 ^ 2 = 2 W1 ^ 2 + W1 - 3 - 2 W1 ^ 2
0 = W1 - 3 Add 3 to both sides
0 + 3 = W1 - 3 + 3
3 = W1
W1 = 3 cm
L1 = 2 W1 = 2 * 3 = 6 cm
To find the dimensions of the first rectangle, let's assume the width of the first rectangle as 'x' cm.
Given that the length of the first rectangle is twice as long as it is wide, the length would be '2x' cm.
The area of the first rectangle is the product of its length and width:
Area of the first rectangle = Length * Width = (2x) * x = 2x^2 cm^2.
Now, let's determine the dimensions of the second rectangle based on the given information.
The length of the second rectangle is 3cm longer than the first one, so it would be (2x + 3) cm.
The width of the second rectangle is 1cm narrower than the first one, so it would be (x - 1) cm.
The area of the second rectangle is the product of its length and width:
Area of the second rectangle = Length * Width = (2x + 3) * (x - 1) cm^2.
Since both rectangles have the same area, we can set up the equation:
2x^2 = (2x + 3)(x - 1).
Expanding the equation:
2x^2 = 2x^2 + 2x - 3x - 3.
Simplifying the equation:
0 = -x - 3.
Rearranging the equation:
x = -3.
Since the width cannot be negative in this context, the dimensions of the first rectangle are not defined. It appears that there is an error or missing information in the problem statement.