Optimization Problems

  1. Among all pairs of real numbers that have a sum of \(33,\) which pair has the largest product?
  2. Among all pairs of real numbers that have a difference of \(42,\) which pair has the smallest product?
  3. If the sum of two positive real numbers is \(83,\) what is the minimum possible value of the sum of their squares?
  4. What is the minimum vertical distance between the parabolas corresponding to the equations \({y = x^2+x+3}\) and \({y = 1+3x-2x^2\,?}\)
  5. You are standing near the edge of a 264-foot high cliff. You hurl a rock over the edge of the cliff with an initial upward velocity of 51 ft/s and soon hear it clack against the hard pebbled ground at the cliff’s base. The altitude of the rock above the ground \(t\) seconds after you hurl it can be modelled by the function \({A(t)=264+51t-16t^2.}\) What is the maximum height (altitude) above the ground that the rock reaches?
  6. Farmer Allison wants to construct a rectangular, fenced-in pen for her herd of sheep. She has 420′ of prefabricated fencing. What’s the maximum possible area of land the pen can enclose?
  7. Farmer Bruce want to build a rectangular, fenced-in pen for his herd of goats along a riverbank; this way he only needs to build three sides of the pen and his goats will be free to frolic in the water. He has 420′ of prefabricated fencing. What’s the maximum possible area of land the pen can enclose?
  8. Farmer Carl plans on building a rectangular, fenced-in pen for his aardvarks and his badgers. The aardvarks and badgers shouldn’t mingle though, obviously, so Carl plans on building another length of fencing through the middle of the pen, parallel to two opposite sides, to separate the aardvarks and badgers into two smaller equal-sized rectangular pens.
  9. Show that the rectangle that encloses the greatest area per a fixed given perimeter is a square.
  10. The vertex of a parabola is the location of its extremum. Show that the \(x\)-coordinate of the vertex of the parabola given by \({y = ax^2+bx+c}\) is \({\tfrac{-b}{2a}.}\)
  11. What are the coordinates of the point on the line \(y = 5-7x\) that is closest to the origin?
  12. Any line that passes through the point \((2,3)\) and has a negative slope will form a triangle with positive \(x\)- and \(y\)-axes. What slope minimizes the area of this triangle?
  13. What are the dimensions, in terms of \(r,\) of the rectangle of largest possible area that can be inscribed in a semicircle of radius \(r\) where one side of the rectangle lies in the diameter of the semicircle.
  14. openstax A rectangle is to be inscribed in the ellipse \(\tfrac{x^2}{4} + \tfrac{y^2}{1} = 1\) with its sides square with the coordinate axis (i.e. it isn’t tilted). What should the dimensions of the rectangle be to maximize its area?
  15. What are the dimensions of an isosceles triangle inscribed in a circle of radius \(r\) (with vertices on the boundary of the circle) that has maximal area?
  16. Suppose one side of a triangle has length \(\ell\) and another side has length \(2\ell\). Show that he maximum possible area of the triangle is \(\ell^2.\)
  17. An open-top box is to be constructed from a square, 42″×42″ piece of cardboard, by cutting small squares from the corners of the cardboard and folding up and taping together the resulting “flaps” to be the sides of the box. What size squares should be cut from the corners to maximize the volume of the resulting box?
  18. A closed-top box is to be constructed from a square, 42″×42″ piece of cardboard, by cutting squares and rectangles from the corners of the cardboard and folding up and taping together the resulting “flaps” to be the sides and lid of the box. What size squares and rectangles should be cut from the corners to maximize the volume of the resulting box?
  19. A 42 ft³ closed-top box is to be manufactured with the following specifications: the length of the base of the box will be twice its width, the top and bottom of the box will be constructed out of carbon steel which costs $23 per square foot, the sides will be constructed out of galvanized steel which costs $17 per square foot, and all the sides will be welded together along the edges. What dimensions will minimize the cost to manufacture the box?
  20. The cost of producing a quantity of plumbuses, measured in thousands of dollars per each gross of plumbus \(x,\) is given by the quadratic function \({C(x) = x^2+5x+6.}\) The average cost-per-plumbus is given by the formula \(\tfrac{C(x)}{x}.\) How many gross of plumbuses minimizes this average cost?

  21. The Coca-Cola Company plans on cutting manufacturing costs. They want to redesign the shape of their 12 fl oz soda can to use as little aluminum as possible. What are the dimensions of a can that will achieve this?

  22. You must cut a 42 inch long length of copper wire into two pieces, bending one piece into a square and the other into a circle. Where do you cut the wire so that the total area enclosed by the square and the circle is maximized? Where do you cut the wire so that the total area enclosed by the square and the circle is minimized?
  23. An ad agency is manufacturing a large wallscape (like a billboard, but on a building) to place on the facade of an office building. It’ll be made out of particleboard, painted white, then have the advertisement printed on it using an industrial printing machine. There’s enough material to make a particleboard with an area of 738 ft². The industrial printing machine can’t print on the entire board though: there must be 2 ft margins on the left and right edges of the ad and 3 ft margins on the upper and lower edges of the ad. The ad agency wants the advertisement on the board to be as large as possible though. What is the maximal possible area of the printed ad?
  24. A water trough, shaped like a big rain gutter, is to be made from a long strip of tin 8 inches wide by taking \(x\) inches on each side of that width (so \(x \lt 4\)) and bending them up at and angle 60° (\(\tfrac{\pi}{3}\) radians). What length \(x\) maximizes the cross sectional area, and thus the amount of water, the trough holds?
  25. Jungic, Menz, Pyke A water trough, shaped like a big rain gutter, is to be made from a long strip of tin 8 in wide by taking 3 in on each side of that width and bending them up at and angle \(\theta\). What angle \(\theta\) maximizes the cross sectional area, and thus the amount of water, the trough holds?
  26. Sonya is floating on a raft 2 km offshore. She hasn’t eaten in a couple of days, and she sees a restaurant on the straight shoreline, 6 km from the point on the shore nearest to her raft. Sonya can row at 3 km/hr and sprint at 5 km/hr. Wanting to get to the restaurant as quickly as possible, towards what point along the beach should she row her raft?
  27. Jungic, Menz, Pyke A storage container is to be made in the form of a right circular cylinder with a volume of 287 ft². Material for the top of the container costs $5 per square foot and material for the sides and base costs $2 per square foot. What dimensions will minimize the total cost of the container?
  28. a circle with a segment missing, and arrow pointing to the conical (paper) cup that the circle could be folded into A sector of a circle can be folded up into a cone. Some cheap water cups are constructed this way: start with a disk of waxed paper, make two cuts to remove a small sector of the disk, and adhere the disk along the two cuts to make a “cup”. (Obviously it’s more complicated than this; you need to leave a little paper tab to glue across the cut, but let’s keep this problem simple.) Suppose the initial circular sector has radius \(r\) and has a subtended angle measuring \(\theta.\) What angle \(\theta\) corresponds to the cup that holds the maximal amount of water?
  29. A 13 foot-tall fence runs along the outside of a building. There is a 4 foot-wide corridor between the fence and building. What’s the shortest ladder that can be laid over the barbed-wire top along the top of the fence and still reach the wall of the building? (asking for a friend)

  30. A Norman window has a shape of a rectangle surmounted by a semicircle — it has three straight sides and one curved side on top, and the diameter of the semi-circular curved side is equal to the width of the rectangle.

    1. If the perimeter of the window is fixed at 12 ft, what are the dimensions that result in the window having maximal area?
    2. If the area of the window is fixed at 10 ft², what are the dimensions that result in the window having minimal perimeter?
  31. openstax Two poles are connected by a wire that is also connected to the ground. The first pole is 20 ft tall and the second pole is 10 ft tall. There is a distance of 30 ft between the two poles. Where should the wire be anchored to the ground to minimize the amount of wire needed?
  32. Take a 8½″×11″ piece of letter paper. Which point along long side of the paper should you fold an opposite corner of the paper to, matching the corner of the paper to that point, to maximize the length of the fold?
  33. Sonya is on the shore of a perfectly circular lake of radius 2 km, sitting near her favorite raft. She hasn’t eaten in a couple of days, and she sees a restaurant directly opposite her across the lake. Sonya can row at 3 km/hr and sprint at 5 km/hr, and she figures that to get to the restaurant as quickly as possible, she could row in a straight line to some point on the shore near the restaurant and then sprint the rest of the distance. Towards what point along the shore of the lake should she row her raft?
  34. Jungic, Menz, Pyke With a straight piece of wire 4 m long, you are to create an equilateral triangle and a square, or either one only. Suppose a piece of length \(x\) meters is bent into triangle and the reminder is bent into a square. Find the value of \(x\) which maximizes the total area of both triangle and square.
  35. Playing fetch with your dog at the beach, you throw the ball into the ocean. It lands and floats 100 ft down the straight shoreline, bobbing in the calm ocean 30 ft off shore. Your dog is smart, and wants to reach that ball as quickly as possible, first running some ways down the shoreline before jumping in the water. If your dog can run at 21 ft/s and can swim at 4 ft/s, how far down the shoreline will it run before jumping in the ocean?

  36. A long steel pipe is being carried down a hallway. The hallway is 15 ft wide but makes an abrupt right-angled turn into a hallway that is only 6 ft wide.

    1. If the pipe has to be carried perfectly level, what is the longest the pipe could possibly be to make this turn?
    2. If the hallways are 12 ft tall and the pipe can be tilted, what is the longest the pipe could possibly be to make this turn?
  37. Siskel and Ebert are going to Movie Night in the Park to watch the latest Batman film. The “screen” is a large white tarp tethered taught between two trees, perfectly vertical, its top 45 feet off the ground and its bottom 5 feet off the ground. The “theater” is a level length of lawn lingering long away from the screen where folks are putting down picnic blankets to claim their seat. Siskel brought pillows and a blanket so they can lay on their backs and pretend they are watching the Batman in bed. Ebert wants to ensure they get the best viewing angle to watch the film so that, from their perspective, the screen appears as large as possible. How far away from the screen should they sit? (Major hint: \(\tfrac{\mathrm{d}}{\mathrm{d}x} \arctan(x) = \tfrac{1}{1+x^2}.\))
  38. Jungic, Menz, Pyke For positive integers \(m\) and \(n,\) what is the maximum possible value of \(f(x) = x^m(1-x)^n?\)
  39. James Stewart If the tangent line at a point \(P\) on the curve \(y = x^3\) intersects the curve again at \(Q\), let \(A\) be the area of the region bounded by the curve and the line segment \(PQ.\) Let \(B\) be the area of the region defined the same way but starting with \(Q\) instead of \(P.\) What is the relationship between \(A\) and \(B?\)
  40. James Stewart An isosceles triangle is circumscribed about a unit circle so that the equal sides meet at the point \((a,0)\) on the \(x\)-axis. Find the value of \(a\) that minimizes the lengths of the equal sides.
  41. James Stewart The line \(y = mx + b\) intersects the parabola \(y = x^2\) in points \(A\) and \(B\). Find the point \(P\) on the arc the parabola that maximizes the area of the triangle \(PAB\).
  42. Farmer Ximena plans on building a rectangular, fenced-in pen for her capybara and her ducks. The capybara and ducks shouldn’t mingle though, obviously, so Ximena plans on building another length of fencing along the diagonal of the pen to separate the capybara and ducks into two smaller equal-sized triangular pens. She has 420′ of prefabricated fencing. What’s the maximum possible area of land the pen can enclose?