In a thin rectangular tin sheet 8x12 sheet, you have to cut a rectangular opening. The area of the opening has to be 45 feet, and its edges have to have an equal distance from the edges of the sheet. Determine this distance.
if the distance from the edges is x,
(8-2x)(12-2x) = 45
x = 3/2
So, the hole is 5x9 with area 45
To determine the distance from the edges of the sheet to the edges of the rectangular opening, we first need to find the dimensions of the rectangular opening.
Let's assume the distance from the edges of the sheet to the edges of the rectangular opening is "x" feet.
Since the area of the rectangular opening is 45 square feet, its length multiplied by its width should equal 45. Let's represent the length and width of the opening as "L" and "W" respectively.
From the given information, we know that the length of the tin sheet is 8 feet and the width is 12 feet. Therefore, the length and width of the rectangular opening can be expressed as:
L = 8 - 2x (subtracting twice the distance from the length)
W = 12 - 2x (subtracting twice the distance from the width)
Now, we can calculate the area of the rectangular opening using the dimensions obtained above:
(L)(W) = 45
Replacing the length and width values:
(8 - 2x)(12 - 2x) = 45
Expanding the equation:
96 - 16x - 24x + 4x^2 = 45
Rearranging the equation:
4x^2 - 40x + 51 = 0
Now, we can solve this quadratic equation to find the value of x.
Using the quadratic formula, x can be calculated as:
x = (-b ± √(b^2 - 4ac)) / (2a)
Substituting the values into the formula:
x = (-(-40) ± √((-40)^2 - 4 * 4 * 51)) / (2 * 4)
x = (40 ± √(1600 - 816)) / 8
x = (40 ± √784) / 8
x = (40 ± 28) / 8
Simplifying:
x1 = (40 + 28) / 8 = 68 / 8 = 8.5
x2 = (40 - 28) / 8 = 12 / 8 = 1.5
Therefore, there are two possible values for "x": 8.5 feet and 1.5 feet.