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.
the minor axis is still 5.5
The major axis is 5.5/cos(arctan .25)
check my thinking.
To find the equation of the ellipse that should be cut from the roof, we need to determine the dimensions of the ellipse.
First, let's find the major axis of the ellipse. The major axis is the diameter of the circular vent pipe, which is given as 5.5 inches.
Next, we need to find the minor axis of the ellipse. The minor axis represents the width of the ellipse, which must accommodate for the slope of the roof. The slope of the roof is given as 1/4.
To determine the width of the ellipse, we divide the diameter of the circular pipe by the slope of the roof:
Width = Diameter / Slope = 5.5 inches / (1/4) = 5.5 inches × (4/1) = 22 inches.
Now we have the dimensions of the ellipse: the major axis is 5.5 inches and the minor axis is 22 inches.
The equation of an ellipse with center (h, k), major axis a, and minor axis b is:
((x - h)^2) / (a^2) + ((y - k)^2) / (b^2) = 1.
Since the circular vent pipe is centered on the ellipse, the center of the ellipse is the same as the center of the pipe.
Therefore, the equation of the ellipse that should be cut from the roof in order to fit the pipe is:
((x - h)^2) / (5.5^2) + ((y - k)^2) / (22^2) = 1.
Note: The values of h and k are not provided in the question, so you will need additional information about the placement of the pipe to determine the exact equation of the ellipse.