A circular vent pipe with a diameter of 8.5 inches is placed on a roof that has a slope of as shown in the figure below. 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?
A)22.89 inches and 4.25 inches
B)9.15 inches and 4.25 inches
C)9.15 inches and 8.50 inches
D)4.25 inches and 3.40 inches
E)8.50 inches and 3.40 inches

Question

A circular vent pipe with a diameter of 8.5 inches is placed on a roof that has a slope of  as shown in the figure below.  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?
A)22.89 inches and 4.25 inches 
B)9.15 inches and 4.25 inches 
C)9.15 inches and 8.50 inches 
D)4.25 inches and 3.40 inches 
E)8.50 inches and 3.40 inches

Quizplus · Accepted Answer

Since the diameter of the circular vent pipe is 8.5 inches, its radius is 4.25 inches. Let's call this value "a". We need to find the other radius, which we'll call "b".
To do this, we need to find the distance from the center of the ellipse (which is also the center of the circle) to the far edge of the ellipse. This can be found using the Pythagorean theorem:
$$
b^2 = a^2 + h^2
$$
where h is the height of the circular pipe above the roof. We're given that the slope of the roof is 11, which means that for every 12 units of horizontal distance, the roof rises by 11 units. We can use this information to find h:
$$
	an(	heta) = \frac{11}{12}
$$
where θ is the angle of the roof. We can use the inverse tangent function to find θ:
$$
	heta = 	an^{-1}\left(\frac{11}{12}ight) \approx 40.44^\circ
$$
Now we can find h using trigonometry:
$$
\cos(	heta) = \frac{h}{12}
$$
$$
h = 12\cos(	heta) \approx 9.15	ext{ inches}
$$
Now we can solve for b:
$$
b^2 = 4.25^2 + 9.15^2 \approx 84.54
$$
$$
b \approx 8.50	ext{ inches}
$$
Therefore, the lengths of the major and minor axes are approximately 9.15 inches and 8.50 inches, respectively. So the answer is C.

A Circular Vent Pipe with a Diameter of 8