Find the dimensions of a right circular cylindrical can with both a top and a bottom that holds 17576 cubic cm and is constructed with the least amount of material possible. Radius of can= cm Height of can= cm Area of Material = cm2Find the point on the line 6x + y = 9 that is closest to the point (74, 73). X = H: y = EEE Note: Instead of minimizing the distance, it is equivalent to minimize the square of the distance which is easier to work with in this case. A rectangular photo frame encloses an area of 700 61712. The top edge of the frame is constructed out of heavier material than the other three sides. The material for the top edge weighs 200 g/cm and the other three sides are made from material weighing 100 g/cm. In this question we will calculate the dimensions of the frame that minimize the total weight. 41151 a) Write an expression in terms of :1: and y forthe total weight of the frame. Weight = ::: g b) Find a constraint that allows you to express 3; in terms of a; 29' = iii c) The values of :1: and y that minimize the total weight are 32min = 55! cm ymin u. cm A construction crew has to dig ditches for a set of pipes that must carry water from point A to both of points B and C {see figure). Points B and C are 4 km apart and point A is 4 km from the midpoint of B and C. They want to dig as little as possible and they believe the best way to do that is to dig from point Ato some point D along the midline a distance h from the midpoint of B and C. They know that they should choose h satisfying 0 g h g 2 but they are not sure exactly what value of h to use. 4km 4km A a) Formulate an expression L(h) for the total length of ditch. L(h) = EEE b) Now calculate the value of h which minimizes the amount of ditch to be dug. If there is more than one such value, give your answer as a comma separated list. h = :=: Social animals that live in groups can spend less time scanning for predators than solitary individuals. However, they waste time fighting with the other group members over the available food. There is some group size at which the net benefit is greatest because the animals spend least time on these unproductive activities, and thus can spend time on feeding, mating, etc. Assume that for a group of size m, the fraction of time spent scanning for predators is 1 and the fraction of time spent fighting with other animals over food is F(:L') : B[a: + 1)6 where A, B are constants and A > BB. Find the size of the group for which the time wasted on scanning plus time wasted on ghting is smallest. 3:: A painting in an art gallery has height h and is hung so its lower edge is a distance d above the eye of an observer (as in the figure). How far from the wall should the observer stand to get the best view? (In other words, where should the observer stand so as to maximize the angle 6 subtended at their eye by the painting?) (Your answer may depend on h and d.) {#hi The frame for a kite is to be made from six pieces of wood. The four exterior pieces have been cut with the lengths indicated in the figure. To maximize the area of the Kite, how long should the diagonal pieces be? List the diagonal pieces lengths in non-decreasing order. Your answers may depend on a and b and assume a