Final answer:
To find the length and breadth of the rectangle, we set up equations for the perimeter and area, solved a quadratic equation, and found that the rectangle's dimensions are either a length of 17 cm and breadth of 4 cm, or the reverse.
Step-by-step explanation:
To solve for the length and breadth of the rectangle when given the perimeter and area, we can use two equations derived from the properties of a rectangle. For a rectangle with length (L) and breadth (B), the perimeter (P) is 2(L + B) and the area (A) is L × B. We are given that the perimeter is 42 cm and the area is 68 cm2.
First, we set up the equation for the perimeter:
P = 2(L + B) = 42
L + B = 21 ... (1)
Next, we set up the equation for the area:
A = L × B = 68 ... (2)
We can now use equation (1) to express B in terms of L:
B = 21 - L ... (3)
Substitute equation (3) into equation (2) to find L:
68 = L × (21 - L)
Solve this quadratic equation by expanding and rearranging it into the standard form and then factoring or using the quadratic formula:
L2 - 21L + 68 = 0
(L - 17)(L - 4) = 0
Therefore, we have two solutions for L, which are L = 17 cm or L = 4 cm. Since both values are positive and valid for the length of a rectangle, one will be the length and the other the breadth.
So, if the length is 17 cm, the breadth (B) using equation (3) would be:
B = 21 - 17 = 4 cm
Alternatively, if the length is 4 cm, the breadth would be:
B = 21 - 4 = 17 cm
The rectangle could have a length of 17 cm and breadth of 4 cm, or vice versa. Both options satisfy the given perimeter and area conditions.