To find the dimensions of a rectangular living room, where we know the perimeter and the area, we can set up two equations based on the formulas for perimeter and area of a rectangle:
1. The perimeter (P) of a rectangle is calculated by the formula P = 2 * length + 2 * width. In our case, the perimeter is 40 meters:
\[ 2 * \text{length} + 2 * \text{width} = 40 \]
\[ \text{length} + \text{width} = 20 \] (by dividing both sides by 2)
2. The area (A) of a rectangle is calculated by the formula A = length * width. We know the area is 100 square meters:
\[ \text{length} * \text{width} = 100 \]
Let's denote the length by L and the width by W. We now have two simultaneous equations:
\[ L + W = 20 \]
\[ L * W = 100 \]
From the first equation, we can express W in terms of L:
\[ W = 20 - L \]
Substituting this into the second equation gives us:
\[ L * (20 - L) = 100 \]
Expanding this, we get a quadratic equation:
\[ 20L - L^2 = 100 \]
\[ L^2 - 20L + 100 = 0 \]
To find the length (L), we can solve the quadratic equation for L. This is a standard form quadratic equation, and it can be factored as follows:
\[ (L - 10)^2 = 0 \]
This means L - 10 = 0, so L = 10. Now we need to find the width (W). Using the earlier substitution:
\[ W = 20 - L = 20 - 10 = 10 \]
So the dimensions of the living room are 10 meters by 10 meters.
