Final answer:
To determine the dimensions of the painting, set up a system of equations using the given area (36 square feet) and perimeter (24 feet). By substituting the perimeter equation into the area equation, we deduce that the painting is a square with dimensions of 6 feet by 6 feet.
Step-by-step explanation:
To find the dimensions of a rectangular painting given the area and the perimeter, we can set up a system of equations based on the formula for the area (A = length × width) and the formula for perimeter (P = 2(length + width)). Since we know the area of the painting is 36 square feet and the perimeter is 24 feet, we can write two equations:
1) Area equation: lw = 36
2) Perimeter equation: 2(l + w) = 24
We can rearrange the perimeter equation to find an expression for w:
l + w = 12
w = 12 - l
Now, substitute this expression for w into the area equation:
l(12 - l) = 36
This leads to a quadratic equation:
l2 - 12l + 36 = 0
Factoring this quadratic equation, we find:
(l - 6)(l - 6) = 0
So, l = 6 feet. And since w = 12 - l, w also equals 6 feet. Thus, the dimensions of the rectangle are 6 feet by 6 feet.