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?
What quantity is it that you want to optimize? Write down an expression for this quantity in terms of some variables (any variables). Use a picture to keep track of what the variables mean.
What quantity constrains your problem? Write down one or more expressions for this constraint quantity in terms of some variables (maybe the same from your picture and optimizing quantity).
Use the constraint expressions to substitute away all but one variable in the optimization expression. That is, write the quantity you want to optimize as a function of a single variable.
Find two numbers exactly 20 apart whose product is as small as possible. For example, if your numbers were 3.5 and 23.5, their product would be 82.25; can you find a pair with smaller product?
A javelina rancher wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. Each pen will be covered with mulch that costs $5 per cubic meter, to a depth of 3 inches. Each pig requires 3 square feet of space. He has 780 feet of fencing available to complete the job. What is the largest number of javalinas the farmer to enclose?
Next door, a vicuña wool grower wants to enclose a rectangular area along the side of an existing wall. Fencing costs $25 per foot. He needs the enclosure to have an area of 300 square feet. What is the least amount of money he can spend to complete the job?
Your construction firm as been contracted to install a rectangular swimming pool with an area of 300 square feet, surrounded on all sides by a wood deck. On three sides, the wood deck must extend 3 feet from the pool. On the fourth side, the deck should extend 7 feet. What dimensions for the pool will minimize the amount of wood needed for the deck?
Oscar has walked about 1 mile along the straight bank of a lake when he notices a puppy drowning. Without a care for his own safety, Oscar swims 1/4 of a mile straight out to save the puppy. Obviously, he wants to get back to his car as quickly as possible. He figures he can swim at a rate of 3 miles per hour and run at a rate of 7 miles per hour. To what point along the bank of the lake should this hero swim?
A Norman window has the shape of a semicircle atop a rectangle so that the diameter of the semicircle is equal to the width of the rectangle. What is the area of the largest possible Norman window with a perimeter of 39 feet?
A wire of length 13 is cut into two pieces which are then bent into the shape of a circle of radius \(r\) and an equilateral triangle with side length \(s\text{.}\) Where should the wire be cut to maximize the combined area of the two shapes?
Consider the region in the \(x\)-\(y\) plane that is bounded by the \(x\)-axis and the function \(f(x) = 25-x^2\text{.}\) Construct a rectangle whose base lies on the \(x\)-axis and is centered at the origin, and whose sides extend vertically until they intersect the curve \(y = 25-x^2\text{.}\) Which such rectangle has the greatest perimeter?
You have been hired to build a rectangular trash can without a lid out of pure gold sheet metal for the new White House bathroom. The material costs $440 per square inch and you have a budget of $2 million. What is the largest amount of trash the trashcan can hold?
