Jamboard Problem 4a (optimization) A soup can in the shape of a right circular cylinder is to be made from two materials. The material for the side of the can costs $0.015 per square inch and the material for the lids costs $0.027 per square inch. Suppose that we desire to construct a can that has a volume of 16 cubic inches. What dimensions minimize the cost of the can? ~15 x] ' IF 1' variables. Use your variables to determine expressions for the volume. surface area, and cost of the can. Determine the total cost function as a function of a single variable. What is the domain on which you should consider this function? Find the absolute minimum cost and the dimensions that produce this value. Write your answers in one complete sentence. Draw a picture of the can and label its dimensions with appropriate I? Jamboard Problem 4b (related rates) A water tank has the shape of an inverted circular cone (point down) with a base of radius 6 feet and a depth of 8 feet. Suppose that water is being pumped into the tank at a constant instantaneous rate of4 cubic feet per minute. 1. Draw a picture of the conical tank. including a sketch of the water level at a point in time when the tank is not yet full. Introduce variables that measure the radius of the water's surface and the water's depth in the tank, and label them on your figure. Say that ris the radius and h, the depth of the water at a given time, 1'. What equation relates the radius and height of the water. and why? Determine an equation that relates the volume of water in the tank at time tto the depth h of the water at that time. Through differentiation, nd an equation that relates the instantaneous rate of change of water volume with respect to time to the instantaneous rate of change of water depth at time t. Find the instantaneous rate at which the water level is rising when the water in the tank is 3 feet deep. When is the water rising most rapidly: at h = 3, h = 4, or it = 5'? Explain how you know