A circular vent pipe with a diameter of 5.5 inches is place on a roof that has a slope of 1/4.

Find an equation of the ellipse that should be cut from the roof so that the pipe will fit. roound parameters to the nearest hundreth.

To find the equation of the ellipse that should be cut from the roof to fit the circular vent pipe, we can follow these steps:

Step 1: Determine the major and minor axes of the ellipse.
The major axis of the ellipse will be the diameter of the circular vent pipe, which is given as 5.5 inches.
The minor axis of the ellipse will be equal to the diameter of the circular vent pipe projected onto the roof. Since the roof has a slope of 1/4, we need to calculate the length of this projected diameter.

To find the length of the projected diameter, imagine a right triangle formed by the circular vent pipe's diameter (5.5 inches), the projected diameter, and the vertical distance from the roof to the circular vent pipe. We can use the Pythagorean theorem to calculate the length of the projected diameter.

Let x be the length of the projected diameter:
x^2 = (5.5/2)^2 + (1/4)^2
x^2 = 15.0625 + 0.0625
x^2 = 15.125
x ≈ √15.125
x ≈ 3.88 inches (rounded to the nearest hundredth)

So, the major axis of the ellipse is 5.5 inches and the minor axis is 3.88 inches.

Step 2: Determine the coordinates of the center of the ellipse.
Since the vent pipe is a circular shape and the ellipse is being cut out of the roof to fit it, the center of the ellipse will coincide with the center of the vent pipe. Therefore, the coordinates of the center of the ellipse will be (0, 0) in the x-y plane.

Step 3: Write the equation of the ellipse in standard form.
The equation of an ellipse with a vertical major axis and center (h, k) can be written in standard form as:
(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1

In this case, the center (h, k) is (0, 0), and the major and minor axes are a = 5.5/2 = 2.75 (rounded to the nearest hundredth) and b = 3.88/2 = 1.94 (rounded to the nearest hundredth):

(x - 0)^2 / (2.75)^2 + (y - 0)^2 / (1.94)^2 = 1

Simplifying, we have the equation of the ellipse:

x^2 / 7.5625 + y^2 / 3.7636 = 1

So, the equation of the ellipse that should be cut from the roof to fit the circular vent pipe is approximately:

x^2 / 7.56 + y^2 / 3.76 = 1