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Let's assume that the rectangle has length 'l' and width 'w'.
The formula for the area of a rectangle is A = l × w, and the formula for the perimeter (P) of a rectangle is P = 2(l + w).
We're given that the area of the rectangle is 6cm², so:
A = l × w = 6cm²
We're also given that the perimeter of the rectangle is 14cm, so:
P = 2(l + w) = 14cm
We can use the second equation to solve for one of the variables in terms of the other:
l + w = 7
l = 7 - w
Substituting this into the equation for the area, we get:
(7 - w) × w = 6
Expanding the left side of the equation, we get:
7w - w² = 6
Rearranging and factoring, we get:
w² - 7w + 6 = 0
This equation can be factored as:
(w - 1)(w - 6) = 0
So, w = 1 or w = 6.
If w = 1, then l = 7 - w = 6, and the perimeter would be:
P = 2(l + w) = 2(6 + 1) = 14
If w = 6, then l = 7 - w = 1, and the perimeter would still be:
P = 2(l + w) = 2(1 + 6) = 14
Therefore, there are two possible solutions: a rectangle with dimensions 1cm × 6cm, and a rectangle with dimensions 6cm × 1cm.
Area = lb =6
P = 2(l + b) =14
Hence l + b = 7
l = 6/b
Substitute
(6/b) + b = 7
Multiply through by 'b'
Hence
6 + b^2 = 7b
b^2 - 7b + 6 = 0
Factor
(b - 6)(b - 1) = 0
Hence
b = 6 & b = 1 However, common sense dictates b = 1 & l = 6
To find the dimensions of a rectangle with an area of 6cm² and a perimeter of 14cm, we can use the following steps:
Let's start by using the formula for the perimeter of a rectangle, which is: P = 2L + 2W, where P is the perimeter, L is the length, and W is the width. Since we know that the perimeter is 14cm, we can write:
14 = 2L + 2W
Next, let's use the formula for the area of a rectangle, which is: A = LW, where A is the area, L is the length, and W is the width. Since we know that the area is 6cm², we can write:
6 = LW
Now we have two equations with two variables (L and W), so we can solve for one variable in terms of the other and substitute into the other equation. Let's solve for L in terms of W from the first equation:
14 - 2W = 2L
L = (14 - 2W) / 2
L = 7 - W
Now we can substitute this expression for L into the second equation:
6 = LW
6 = W(7 - W)
Simplifying this equation gives us a quadratic equation:
6 = 7W - W²
W² - 7W + 6 = 0
We can solve this quadratic equation by factoring:
(W - 6)(W - 1) = 0
This gives us two possible values for W: W = 6 or W = 1.
If we substitute these values back into the equation for L, we get:
L = 7 - W
For W = 6: L = 1
For W = 1: L = 6
So the dimensions of the rectangle could be 1cm by 6cm or 6cm by 1cm, depending on how the length and width are defined.
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