Turning a stated problem into one function of one variable to optimise

0:00Here is a sentence. A sheet of card, three metres by eight, has equal squares cut from its corners, and the sides are folded up to make an open box. What cut gives the greatest volume? There is nothing in that sentence to differentiate. That is not a joke. Every method you have for finding a largest value starts with a rule of one variable, and there is no rule of one variable anywhere in what I just read out.

0:32The work of these problems is almost entirely the work of building one. Watch what actually happens. The description gives you two unknowns: the cut, and the shape of the base it leaves. It gives you a quantity to be made large, the volume. And it gives you a second relation, hidden in the words 'folded up', that ties the base to the cut. Use that relation to get rid of one unknown, and you are left with a rule of one variable. Only then does any of the calculus apply.

1:08So this video is about the part before the first derivative, because that is the part that is actually hard, and it is the part that gets one line of explanation and then never gets mentioned again. Four moves, named once, and I will point back at them all the way through. One: name what is to be made largest or smallest. Call that the objective. Two: write it in terms of the unknowns the description gives you. There will usually be two.

1:42Three: find the second relation that ties those two together, and use it to eliminate one. Call that relation the constraint. Four: say what range the surviving variable is allowed to take. After those four, and only after them, you have a rule of one variable on a stretch, and everything you already know applies to it unchanged. Notice which of the four usually gets the attention. Not these. The fifth step, the one that is already familiar.

2:16Two of these four are where the mistakes live, and I want to spend the video on those two: where the constraint comes from, and what the range is worth. First, the constraint, because students often read it as extra information they have been kindly given. It is not extra. It is the only reason the question has an answer. There are three places to look for it. A stated total: two numbers add to fifteen. A shape: a cylinder sits inside a cone, so the cone's slope ties the cylinder's height to its radius. Or a stated dependence: the price a maker gets falls as the number sold rises.

3:03Different dressings, one job. Each of them turns two unknowns into one. And when you cannot find it, you are not stuck on the calculus. You are stuck on the reading, and no amount of differentiating will help. Take the smallest problem of this kind, so the machinery is visible. Two positive numbers add to fifteen. Make the sum of their squares as small as possible. Objective: the sum of the squares. Unknowns: the two numbers. Constraint: they add to fifteen. So the second number is fifteen less the first, and the whole thing collapses to one variable.

3:48The checker did not solve this. It laid a ladder of inputs across the stretch from nought to fifteen, worked out the sum of squares at every rung, kept the best one, and then refined around it, over and over. No rate was taken anywhere. It came back with seven and a half, and a least value of a hundred and twelve and a half. Then it ran the working rule you were taught, on the same problem, and got the same answer. Good. That is what agreement between two different methods looks like, and it is worth having.

4:26But look at how much of that was calculus. One line. Everything before it was translation. Now let me show you what the constraint is actually worth, because this is measurable and I have not seen anyone measure it. Take the same objective, the sum of two squares, and delete the relation. Two unknowns, nothing tying them, make the sum of squares small. The checker put that to a grid over both unknowns, and then refined the grid and did it again.

5:01Both times it came back with the same thing: the least value is nought, at nought and nought. Of course it is. With nothing tying them, you just make both numbers as small as you are allowed to, and the answer says nothing at all about the question you were asked. Tie them by their sum, and the least value is a hundred and twelve and a half at seven and a half.

5:27So the constraint does not narrow down an answer that was already there. Before the constraint, there was no answer. That is a different thing, and it is why 'find the relation' is the whole job. The second place to look is a shape, and the classic is a cylinder standing inside a cone. The cone is fixed. The cylinder is not: make it wider and it gets shorter, because the cone's slope pushes its top down. Make it narrower and it gets taller. You want the greatest curved surface.

6:04Two unknowns, the cylinder's radius and its height. The constraint is not stated in words anywhere. It is in the figure: the triangle cut off above the cylinder is similar to the whole cone's triangle, and that similarity ties the height to the radius. That is the move. Once the height is written in terms of the radius, the surface is a rule of the radius alone. With a cone of radius six, the checker found the greatest value at a cylinder radius of three, exactly half the cone's, with radius times height equal to eighteen.

6:47And notice what got dropped along the way. The curved surface has a constant multiple in front of it. It was thrown away before any of this. I will come back to why that is allowed, because it is not obviously allowed. Now the fourth move, the range, which almost every worked solution leaves implicit. Back to the box. Cut a square of side x from each corner of a sheet and fold up. The base sides are the sheet's sides less twice the cut, so the volume is the cut times the two shortened sides.

7:26That is a cubic. Its rate vanishes at two inputs, not one. The checker built twelve different sheets and walked every one of them, over a stretch wide enough to hold both turning points. All twelve have exactly two. And in all twelve, exactly one of the two is a cut the description allows. The other one is always the larger cut, and what the arithmetic reports there is never a volume at all. On the eight oblong sheets it comes out negative. On the four square ones it comes out nought.

8:03So the range is not a formality you add at the end. Half the critical points these problems produce are not answers, and nothing in the algebra tells you which half. Let me write one of them out, because the usual explanation for throwing away the second root is a shrug. The three by eight sheet. Turning points at two thirds and at three. Take the cut of three. The shorter side of the base is three less twice three, which is minus three.

8:37There is no box. You have cut past the middle of the sheet from both sides and the two cuts have gone through each other. That is the reason, and it takes one sentence. What the arithmetic reports at that cut is minus eighteen, which is a number and not a volume. At two thirds, the volume is two hundred over twenty-seven, and that is the answer. Say the reason out loud. A root you discard without a reason is a root you will keep next time.

9:11Now the move I promised to come back to, and the one I think is genuinely mis-taught. Twice in this material, the thing that gets optimised is not the thing that was asked for. A distance question is answered by minimising the square of the distance. A surface question throws away a constant multiple. The usual justification, when there is one, is that squares are harmless. They are not. Here is the real rule, and it is worth learning in this form: you may put anything in the way of your objective, so long as that thing only RISES across the values your objective actually takes.

9:56Every word of that is load-bearing. Only rises. Across the values it actually takes. If the thing in the way only rises, then it keeps every comparison the same way round, and a comparison is all a largest value ever was. The checker took that seriously. It put squaring in the way of all six worked problems, and measured two things separately: whether squaring only rises across the values that problem takes, and whether the answer stayed where it was.

10:32Six times out of six, the two readings agreed with each other. Which would be a dull result if they were both always yes. They are not both always yes. Five of the six take values of one sign only. Squaring rises across those, and the answer does not move. The sixth is the profit problem. A maker's profit, revenue less cost, as a rule of the number of units sold.

11:03That profit runs from six hundred in the red, at the largest output, up to seventy-six in the black at two hundred and forty units. Which is the answer. Square it. Squaring does not only rise across values that run from minus six hundred to seventy-six, because a large negative number has a large square. And the answer moves. The greatest SQUARE of the profit is at the largest output of all, where the maker loses the most money possible.

11:37So a reader who reached for the square here because it looked easier would have maximised the size of the loss, and the working would look perfectly correct all the way down. That is why the rule is 'only rises across the values it takes', and not 'squares are harmless'. Now run the move the right way, on the problem it is meant for. A point runs along a curve. Which point is nearest to a fixed point off the curve?

12:09The distance has a square root in it and it is horrible to work with. The squared distance has no root in it and is easy. The checker built them as two separate rules, and never derived one from the other. It found the least value of each by comparison alone. Both least values sit at the same input. The squared distance is five there; the distance itself is the square root of five.

12:39Which is the answer the question asked for, and not the one that was minimised. Read the last line of the question again before you write the last line of your answer. And notice this problem has no ends at all. The point runs along the whole curve. So the candidate list has exactly one member, and the last step of the working rule, pick the largest and smallest of the values you obtained, has nothing to compare it with.

13:12Back to the simplest constraint, a stated total, because there is a pattern in it worth having. Two numbers add to a fixed total. Make one to some power, times the other to some other power, as large as possible. The checker built fifty of these. Ten pairs of powers, on five different totals. Every one of them answered by comparison alone, no rates anywhere. In all fifty, the total gets split in the ratio of the two powers.

13:45That is why two questions that look identical get different answers. With powers one and three, sixty splits into fifteen and forty-five, a quarter and three quarters. With powers two and five, thirty-five splits into ten and twenty-five, two sevenths and five sevenths. And with equal powers it splits down the middle, which is the case everyone remembers and the reason the others feel surprising. A word on the last step, since we are here.

14:20Once you have your one-variable rule and your critical point, you finish by deciding what kind of point it is. There are two ways, and the choice between them is a look at the expression in front of you, not a habit. On four of the six worked problems the second rate is a plain constant. Four, four, minus four, and minus a fiftieth. When that happens, the second test is one line and you should use it.

14:50On the box it depends on the cut, and on the squared distance it depends on the input, so the second test costs you an evaluation rather than a glance. And then there is the trapezium, whose height is a square root. The checker measured its second rate at three inputs and they did not agree. That is the half page of algebra you have seen in that worked solution.

15:17When the first rate is a quotient with a root in it, look at the sign on each side instead. It is the cheaper finisher and it is just as sound. Something these questions do that catches people out: quite often the answer is not a number. The rectangle of greatest area inside a fixed circle. The checker found it, and then compared its two sides: they come out equal. The answer is a shape. It is a square.

15:49The greatest cone that fits in a sphere. Its volume is eight twenty-sevenths of the sphere's, and the circle constant cancels before any of it gets worked out. The answer is a ratio. The two numbers with a fixed sum. The answer is a pair. And many of these questions ask you to PROVE a stated relation, where the answer is an argument and stopping at the critical value answers nothing.

16:19Four different shapes of answer, one shape of question. The critical value is where the work stops being hard, not where it stops. Here is one that is worth being honest about rather than tidying away. A profit is handed to you as a rule of the number of units made, and you are asked for the number that maximises it. It has exactly one turning point. It is at minus two units, and the profit there is a hundred and thirteen.

16:52As arithmetic, that is right and it is the answer. As a statement about a manufacturer, it is not admissible, because no count of units is negative. Ask the same question over the counts that actually exist, and the greatest profit sits at the very start: make none at all. The answer is at an end of the range, not at a turning point. The honest response is to give the number and name the difficulty. A negative answer does not always mean you made a sign error. Sometimes it means the model does not fit the thing it was built for.

17:35There is a run of questions that look like eight different puzzles: a rectangle in a circle, a cone in a sphere, a cylinder in a sphere, a cylinder in a cone, and so on. They are one puzzle. Eliminate using the fixed quantity, whatever it is. Then optimise a SQUARE, so that no root survives to be differentiated. The checker worked four of them that way, all in a unit that makes the constants vanish.

18:06The rectangle of greatest area in a circle has half-width one over the square root of two, which makes it a square. The greatest cone in a sphere of radius one has height four thirds. The greatest cylinder in that sphere has height two over the square root of three. And the greatest cylinder in a cone has height exactly a third of the cone's, whatever the cone is. One pattern, four answers. If you have done one of these properly, you have done all of them.

18:40A few more, worked rather than quoted. A square sheet of side eighteen, cut into a box: the cut is three. A sheet forty-five by twenty-four: the cut is five. Its other turning point is at eighteen, which would leave the shorter side of the base at twenty-four less thirty-six. Minus twelve. Not a box. The nearest point of a given parabola to a fixed point, offered as four options. Written in the horizontal coordinate it is a mess. Written in the VERTICAL coordinate it is an ordinary quadratic with one turning point, at four, and the horizontal coordinate there is twice the square root of two.

19:27The substitution that made it easy is the whole of the work, and the four options are decoration. And a costing problem: a tank of fixed depth and fixed volume, cheapest to build. Both constraints together fix the base area, so the only thing left to choose is the base's SHAPE. It is cheapest square, at a thousand. That one is a constraint problem wearing a coat. Once you see that the depth and volume together leave you one free choice, it is over.

20:04So: four moves, and the calculus is the fourth. Name the objective. Write it in the unknowns. Find the relation that ties them and eliminate one. State the range. Two things to carry away that are not in the usual telling. The constraint is not extra information. Without it the same objective has no answer at all, and the checker measured that rather than asserting it. And optimising something easier is legitimate exactly when the thing you put in the way only rises across the values your objective takes. On five of six worked problems squaring qualifies. On the profit it does not, and it hands you the biggest loss available.

20:54The range is part of the problem. Twelve sheets, twelve rejected cuts, every one of them the larger root, and every one of them a volume that is negative or nought. Draw the figure. Find the relation. Say the range. Then differentiate.