You must cover a hole that is elliptical in shape. You measure the major and minor axes and find them to be 6 ft. and 4.5 ft. respectively. If you want a 6 inch overhang around the hole, how much covering will you need? (Round to the tenths place.)
area of ellipse is πab where a and b are the semi-axes.
so the area of our expanded ellipse is π(3+.5)(2.25+.5) = 30.2 ft^2
thank you
To calculate the amount of covering needed, we can use the formula for the area of an ellipse:
Area = π * (major axis / 2) * (minor axis / 2)
Given:
Major axis = 6 ft.
Minor axis = 4.5 ft.
Plugging these values into the formula:
Area = π * (6 / 2) * (4.5 / 2)
Area = π * 3 * 2.25
Area ≈ 21.2 ft²
To find the amount of covering needed, we need to add the desired overhang to both sides:
Total Covering = Area + (2 * overhang)
Given:
Overhang = 6 inches = 0.5 ft.
Plugging these values into the equation:
Total Covering = 21.2 + (2 * 0.5)
Total Covering = 21.2 + 1
Total Covering ≈ 22.2 ft²
So, you will need approximately 22.2 square feet of covering.
To determine the amount of covering needed, you first need to calculate the perimeter of the elliptical hole.
The formula for the perimeter of an ellipse is given by:
P = 2π * sqrt((a^2 + b^2) / 2)
Where P represents the perimeter, a is the length of the semi-major axis, and b is the length of the semi-minor axis.
Given that the major and minor axes are 6 ft. and 4.5 ft. respectively, the semi-major axis (a) would be half of the major axis (6 ft. / 2 = 3 ft.), and the semi-minor axis (b) would be half of the minor axis (4.5 ft. / 2 = 2.25 ft.).
So, substituting these values into the formula, we have:
P = 2π * sqrt((3^2 + 2.25^2) / 2)
P = 2π * sqrt((9 + 5.0625) / 2)
P = 2π * sqrt(14.0625 / 2)
P = 2π * sqrt(7.03125)
Now, calculate the value inside the square root:
sqrt(7.03125) ≈ 2.649
Therefore, the perimeter P is approximately:
P ≈ 2π * 2.649
Using the approximation π ≈ 3.14, we have:
P ≈ 2 * 3.14 * 2.649
P ≈ 16.64 ft
Now, to determine the total amount of covering needed, add the overhang of 6 inches (0.5 ft) to the perimeter:
Total covering needed = P + overhang
Total covering needed ≈ 16.64 ft + 0.5 ft
Total covering needed ≈ 17.14 ft
Therefore, you will need approximately 17.1 feet of covering to cover the elliptical hole with a 6-inch overhang.