an artist intends to paint a 60-square-foot mural on a large wall that is 18 ft wide and 11 ft tall. find the dimensions of the mural if the artist leaves a border of uniform width around it.
Dimension of mural:
length = 18-2x
width = 11-2x
(18-2x)(11-2x) = 60 , where x < 5.5
expand, set up like a standard quadratic equation and solve for x,
make sure to reject the x where x > 5.5
Hint: it factors nicely
as you would have seen if you had read my response to you urgent post earlier this morning ...
To find the dimensions of the mural, we need to subtract the width of the border from the width and height of the wall.
Let's assume the width of the border is represented by "x".
The dimensions of the mural would be:
Width of the mural: 18 ft - 2x ft (subtracting 2 times the width of the border as it is uniform on both sides)
Height of the mural: 11 ft - 2x ft (subtracting 2 times the width of the border as it is uniform on both sides)
Since the mural is intended to be 60 square feet, we can use the area formula:
Area = Width × Height
So, we have the equation:
60 sq ft = (18 ft - 2x ft) × (11 ft - 2x ft)
Now, we can solve for "x" by rearranging the equation and simplifying:
60 sq ft = (18 ft - 2x ft) × (11 ft - 2x ft)
60 sq ft = (198 ft^2 - 36x ft^2 - 22x ft^2 + 4x^2 ft^2)
60 sq ft = (198 ft^2 - 58x ft^2 + 4x^2 ft^2)
0 = 198 ft^2 - 58x ft^2 + 4x^2 ft^2 - 60 sq ft
0 = 4x^2 ft^2 - 58x ft^2 + 138 ft^2
Now we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. Let's use factoring by finding two numbers that multiply to give 138 and add to give -58:
0 = (2x - 3)(2x - 46)
Setting each factor equal to zero:
2x - 3 = 0 or 2x - 46 = 0
Solving for "x" in each equation:
2x = 3 or 2x = 46
x = 3/2 or x = 46/2
x = 1.5 ft or x = 23 ft
Since the width cannot be negative and a mural with a 23 ft border seems impractical, we can conclude that the width of the border is 1.5 ft.
Now, we can calculate the dimensions of the mural:
Width of the mural = 18 ft - 2x ft
Width of the mural = 18 ft - 2(1.5 ft)
Width of the mural = 18 ft - 3 ft
Width of the mural = 15 ft
Height of the mural = 11 ft - 2x ft
Height of the mural = 11 ft - 2(1.5 ft)
Height of the mural = 11 ft - 3 ft
Height of the mural = 8 ft
Therefore, the dimensions of the mural with a 1.5 ft border are 15 ft (width) and 8 ft (height).
To find the dimensions of the mural, we need to subtract the dimensions of the border from the overall dimensions of the wall.
Let's say the width of the border is represented by 'w'. Since the border is uniform on all sides, the dimensions of the mural will be reduced by '2w' in width and '2w' in height.
The overall dimensions of the mural (excluding the border) will be the inner dimensions of the wall, which can be calculated as follows:
Width of mural: 18 ft - (2w)
Height of mural: 11 ft - (2w)
Now, we know that the area of the mural is 60 square feet, so we can write the equation:
(18 ft - 2w) * (11 ft - 2w) = 60 square feet
Now, let's solve this equation to find the value of 'w':
(18 - 2w)(11 - 2w) = 60
198 - 36w - 22w + 4w^2 = 60
4w^2 - 58w +138 = 60
4w^2 - 58w +78 = 0
We can solve this equation by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:
w = (-b ± √(b^2 - 4ac)) / (2a)
Plug in the values:
w = (-(-58) ± √((-58)^2 - 4*4*78)) / (2*4)
w = (58 ± √(3364 - 1248)) / 8
w = (58 ± √(2116)) / 8
w = (58 ± 46) / 8
w = (58 + 46) / 8 or w = (58 - 46) / 8
w = 104 / 8 or w = 12 / 8
w = 13 or w = 1.5
Since the width of the border cannot be negative and should also be less than half the width of the wall, we can discard 13 as a solution and consider w = 1.5 ft as the width of the border.
Now, let's calculate the dimension of the mural:
Width of mural = 18 ft - (2 * 1.5 ft) = 15 ft
Height of mural = 11 ft - (2 * 1.5 ft) = 8 ft
Therefore, the dimensions of the mural with a 1.5 ft border would be 15 ft by 8 ft.