To find the dimensions of the mural with a uniform border, we need to subtract the border width from the dimensions of the wall.
Given:
Wall dimensions: 18' x 11'
Mural area: 60 square feet
Let's assume the border width is represented by 'x'.
The area of the mural, including the border, is (18 - 2x) * (11 - 2x). This represents the length and width of the mural, considering the border width on all sides.
We are given that the mural area is 60 square feet. So, we have the equation:
(18 - 2x) * (11 - 2x) = 60
To solve for 'x', we can expand the equation:
198 - 36x - 22x + 4x^2 = 60
Rearranging the equation:
4x^2 - 58x + 138 = 0
This is a quadratic equation that can be factored or solved using the quadratic formula. Factoring this equation may not yield whole number solutions.
Alternatively, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 4, b = -58, and c = 138.
Calculating the discriminant (b^2 - 4ac), we get:
(-58)^2 - 4(4)(138) = 3364 - 2208 = 1156
Since the discriminant is positive, we have two solutions:
x = (-(-58) ± √1156) / (2 * 4)
x = (58 ± 34) / 8
Therefore, the possible values for 'x' are:
x₁ = (58 + 34) / 8 = 92 / 8 = 11.5
x₂ = (58 - 34) / 8 = 24 / 8 = 3
Since 'x' represents the border width, a width of 11.5 feet would not make practical sense. Therefore, we can choose 'x' as 3 feet, which is a more reasonable border width.
Now, to find the dimensions of the mural, we substitute 'x' with 3 in the equation:
Mural length = 18 - 2(3) = 18 - 6 = 12 feet
Mural width = 11 - 2(3) = 11 - 6 = 5 feet
Therefore, the dimensions of the mural with a uniform border of 3 feet are 12 feet by 5 feet.