Chapter 10.4, Problem 45ES

As illustrated in the accompanying figure, a carpenter needs to cut an elliptical hole in a sloped roof through which a circular vent pipe of diameter D is to be inserted vertically. The carpenter wants to draw the out- line of the hole on the roof using a pencil, two tacks, and a piece of string (as in Figure 10.4.3b). The center point of the ellipse is known, and common sense suggests that its major axis must be perpendicular to the drip line of the roof. The carpenter needs to determine the length L of the string and the distance T between a tack and the center point. The architect’s plans show that the pitch of the roof is p (pitch $=$ rise over run; see the accompanying figure). Find T and L in terms of D and p.