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Question 607295: A circular vent pip with a diameter of 6.5 inches is placed on a roof that has a slope of 5/8. The intersection of the vent pipe and the roof is an ellipse. To the nearest hundredth of an inch, what are the lengths of the major and minor axes?
Answer by Nihal@SriLanka(22)   (Show Source):
You can put this solution on YOUR website! Consider the cross-section of the vent pipe going through the point at the highest point on the roof where the vent pipe meets the roof. This is a circle with diameter 6.5 inches. The intersection of the vent pipe and the roof is a projection of this circle on the roof at an angle equal to the slope of the roof.
This projection is of the shape of an ellipse whose major axis is the projection of the diameter of the circular cross-section going through the topmost point referred to above. This diameter, its projection on the roof (which is the major axis) and part of the side of the pipe forms a right-angle triangle whose hypotenuse is the major axis of the ellipse. The tangent of the angle between the hypotenuse and the diameter whose projection it is, is given as 5/8.
Hence length of major axis divided by the length of the diameter is equivalent to the secant of the angle whose tangent is 5/8 which is equivalent to square root of 89 divided by 8.
Hence length of major axis = 6.5 (square root of 89)/ 8 inches which can be easily worked out to the required no. of decimal places either manually or using a calculator.
The minor axis of the ellipse is the projection on the roof of the diameter of the circle which is perpendicular to the diameter considered in the above discussion. You can easily visualize that it is also a diameter of the vent pipe and therefore its length happens to be 6.5 inches.
For any clarification of this answer you may email sumanapala@gmail.com.
Note : The forgoing analysis assumes that the vent pipe is fixed vertically above the roof.
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