A circular vent pipe with a diameter of 5.5 inches is placed 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. Round parameters to the nearest hundredth.
To find the equation of the ellipse that should be cut from the roof, we need to determine the major and minor axes lengths of the ellipse.
First, let's find the major axis length:
The diameter of the circular vent pipe is given as 5.5 inches. Since the vent pipe will be perpendicular to the roof, the major axis of the ellipse should be equal to the diameter. Therefore, the major axis length is 5.5 inches.
Next, let's find the minor axis length:
The slope of the roof is given as 1/4. This means that for every 4 inches horizontally, the roof rises by 1 inch vertically. We can represent this as a right triangle with a horizontal base of 4 inches and a vertical height of 1 inch. Using the Pythagorean theorem, we can calculate the hypotenuse, which represents the length of the minor axis of the ellipse:
h^2 = 4^2 + 1^2
h^2 = 16 + 1
h^2 = 17
Now, the hypotenuse (h) represents the minor axis of the ellipse. However, since the minor axis of the ellipse represents the width, we need to divide h by 2 to get the radius:
r = h/2
r = √17 / 2
Therefore, the radius of the minor axis is (√17) / 2 inches.
Now that we have the major axis length (5.5 inches) and the radius of the minor axis (√17 / 2 inches), we can write the equation of the ellipse in the standard form:
(x^2 / a^2) + (y^2 / b^2) = 1
where a is the semi-major axis (half of the major axis) and b is the semi-minor axis (half of the minor axis).
Plugging in the values we found:
(a = 5.5 / 2 = 2.75) and (b = (√17 / 2) / 2 = √17 / 4)
(x^2 / (2.75)^2) + (y^2 / (√17/4)^2) = 1
Simplifying further, and rounding to the nearest hundredth:
(x^2 / 7.56) + (y^2 / 0.53) = 1
Therefore, the equation of the ellipse that should be cut from the roof so that the circular vent pipe will fit is approximately:
(x^2 / 7.56) + (y^2 / 0.53) = 1