Solve these problems:

FIGURE FOR PROBLEM 1010. A paper drinking cup filled with water has the shape of a cone with height h and semivertical angle 0. (See the figure.) A ball is placed carefully in the cup, thereby displacing some of the water and making it overflow. What is the radius of the ball that causes the greatest volume of water to spill out of the cup?4. Archimedes' Principle states that the buoyant force on an object partially or fully submerged in a fluid is equal to the weight of the fluid that the object displaces. Thus, for an object of density po floating partly submerged in a fluid of density py, the buoyant force is given by F = prg , A( y) dy, where g is the acceleration due to gravity and A(y) is the area of a typical cross-section of the object (see the figure). The weight of the object is given by W = pog , A(y)dy (a) Show that the percentage of the volume of the object above the surface of the liquid is p/ - Po 100 Pr (b) The density of ice is 917 kg/m' and the density of seawater is 1030 kg/m'. What percent- age of the volume of an iceberg is above water? (c) An ice cube floats in a glass filled to the brim with water. Does the water overflow when the ice melts? (d) A sphere of radius 0.4 m and having negligible weight is floating in a large freshwater lake. How much work is required to completely submerge the sphere? The density of the water is 1000 kg/mlY=L-A 7=0 FIGURE FOR PROBLEM 45. Water in an open bowl evaporates at a rate proportional to the area of the surface of the water. (This means that the rate of decrease of the volume is proportional to the area of the surface.) Show that the depth of the water decreases at a constant rate, regardless of the shape of the bowl.6. A sphere of radius I overlaps a smaller sphere of radius / in such a way that their intersection is a circle of radius r. (In other words, they intersect in a great circle of the small sphere.) Find r so that the volume inside the small sphere and outside the large sphere is as large as possible