Please answer the questions attached below showing all your work. Thank you!

2. Suppose we are planing to make a taco from a round tortilla with diameter 8 inches by bending the tortilla so that it is shaped as if it is partially wrapped around a circular cylinder. We will ll the tortilla to the edge with meat, cheese and other ingredients. Our problem is to decide how to curve the tortilla in order to maximize the volume of food it can hold. (a) We start by placing a circular cylinder of radius 7' along a diameter of the tortilla and folding the tortilla around the cylinder. Let x represent the distance from the center of the tortilla to a point P on the diameter. Show that the cross-sectional area of the lled taco in the plane P perpendicular to the axis of the cylinder is A(:r) = n/ 16 $2 %T2 sin (EV 16 $2) and write an expression for the volume of the lled taco. (b) Using the graph generated by a computer algebra system, determine the approx imate value of 7' that maximizes the volume of the taco. 3. Archimedes' Principle states that the buoyant force on an object partially or fully submerged in a uid is equal to the weight of the uid that the object displaces. For an object of density p0 oating partly submerged in a uid of density pf, the buoyant force is given by F = pig ffh 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. The weight of the object is given by W = 909 If? A(y)dy- (a) Show that the percentage of the volume of the object above above the surface of the liquid is 100W p\". Pf (b) The density of ice is 917 kg/m3 and the density of seawater is 1030 kg / m3. What percentage of the vlue of an iceberg is abo ve water? (c) An ice cube oats in a glass lled to the brim with water. Does the water overow when the ice melts