Ski Cabin Problem

Boris and Natasha are planning to build an A-frame ski hut. In addition to having a ground floor, they also plan to build a sleeping loft above.

This plan is a view of the cabin from one end (it is not to scale). The walls and roof extend out perpendicular to this page. The ground floor BC is 30 feet across; the height of the apex A above BC is 25 feet. DE represents both the ceiling of the ground floor room and the floor of the sleeping loft (for simplicity in roughing out our ideas, we are ignoring for now the thickness of the floor of the sleeping loft).

Boris proposes to Natasha that the ceiling to be 10 feet high.

a) With this plan, what is the width DE of the sleeping loft?

b) Boris also likes to be able to stand up in the loft. He is 6 feet tall. For what portion of the loft will a 6-foot person be able to stand erect?

c) Natasha is 5 feet tall. For what portion of the sleeping loft will she be able to stand erect?

Natasha would like to install a large solar panel on the ski hut roof. With this in mind, she thinks a better plan than an A-frame would be a shed roof on one side and a vertical wall on the other. So the end picture would look like the second figure.

d) If the length BC, the height of A above BC, and the ceiling height are kept the same, how will the width DE of the sleeping loft change from that of Boris's plan? How will the portion of the sleeping loft that is 6 feet tall (or more) change?

e) Natasha is still refining her design. In her new plan, which still uses the second figure, she is considering a base BC still 30 feet wide, but she would like the sleeping loft to be changed so that the width DE is now 20 feet wide. With the windows in the wall, she feels that an 8 foot ceiling on the ground floor is enough. With this new plan, how far is A above the ground level BC?

f) Natasha is 5 feet tall. For what portion of the sleeping loft will she be able to stand in this new plan?

g) Boris agrees with Natasha's plan, but wants to add some more room on the side for a bathroom downstairs and a closet upstairs. The plan looks like triangle AFC in the figure. If the width BF of the bathroom is 12 feet, what is the distance DG at the base of the closet?