$11\mathrm{.}75...$ Flying Buttress. $($a$)$ A symmetric building has a roof sloping upward at above the horizontal on each side. If each side of the uniform roof weighs $10,000$ N$,$ find the horizontal force that this roof exerts at the top of the wall, which tends to push out the walls. Which type of building would be more in danger of collapsing: one with tall walls or one with short walls? Explain. $($b$)$ As you saw in part $($a$),$ tall walls are in danger of collapsing from the weight of the roof. This problem plagued the ancient builders of large structures. A solution used in the great Gothic cathedrals during the $1200$s was the flying buttress, a stone support running between the walls and the ground that helped to hold in the walls. A Gothic church has a uniform roof weighing a total of $20,000$ N and rising at above the horizontal at each wall. The walls are $40$ m tall, and a flying buttress meets each wall $10$ m below the base of the roof. What horizontal force must this flying buttress apply to the wall?