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### Course: AP®︎/College Calculus BC>Unit 5

Lesson 11: Solving optimization problems

# Optimization: area of triangle & square (Part 2)

Sal constructs an equilateral triangle & a square whose bases are 100m together, such that their area is the smallest possible. Created by Sal Khan.

## Want to join the conversation?

• If the question had been to maximize combined areas. The answer can't be achieved by finding critical points since the only existing one is a minimum. Does this always mean that there is no way to maximize both areas and therefore one of the figures will have a higher Area/perimeter ratio and therefore all the 100 meter cord should be use to create that figure? I have done the ratio Area/Perimeter of both figures. Being Asq= Area of square; Atr= Area of triangle; Psq= Perimeter of square; Ptr= Perimeter of triangle and X= size of the side of square. These ratios (Asq/Psq = 1/4 X and Atr/Ptr~1/5X) tell me that the solution to optimize areas is to only do a square with all the 100 meter cord. Is there any better way to get to this solution using derivatives?
• We know that the entire x function concave up because the second derivation is always positive. Coincidentally, x increases the further you are away from the critical point (this can be proved with limits as x approaches infinity or something.) The maximum depends on your constraints--if there were no constraints, then the answer would practically be infinity. But because we are constrained to 0 < x < 100 (we have to cut the wire, which is why x never equals 0 or 100), then we can take the lim x --> 0 (because 0 is further away from the critical point than 100), and we get 625. That is our maximum value.
• at 1.10 why is derivative of sqrt3*x^2/36 is equal to sqrt3*x/18?
why isn`t quotient rule for derivative applied in here?
• The quotient rule is not applicable here because the denominator doesn't involve x. The only term involving x we have is x² ("up top" in the numerator), its coefficient being √3/36.
• Given the exact same problem, how would i solve which it for the maximum area?
I mean when you derivate it gives a minimum point.
• Unless you impose a closed interval of allowed lengths, you don't have a maximum area. In this case, there is a physical measurement the requires that 0 ≤ x ≤ 100
This is a closed interval, so what we do is check the value of the original function, not the derivative, at the two endpoints. The absolute max will be whichever of these gives us the greater area.
In this case, if x = 0 it means all of your material went into building the rectangle, thus you have no triangle. If x=100, then all of the material went into building the triangle and you have no rectangle.
If you check the two endpoints, it turns out you have an absolute max at x=0, that is if you only built a rectangle.

If you specify that you cannot use all of the material to build one structure, that you must have both, then you have an open interval. In that case there is no maximum and no way to maximize the enclosed area.
• At which side of the cut string would you use for which shape? Or would it not matter?
• x is equal to the amount of string used in the triangle, and 100 minus x (the remainder of the string) is used to make the square! Yay math!
• This is a question about some of the optimization exercises; they say "The trick here is to recognize that the x value that will maximize the value of S(x) is the same as the x value that maximizes A(x)" in the hints, when S(x) = (A(x))^2. How does that work?

An example is when A(x) = 4xsqrt(r^2-x^2) and S(x) = 16x^2(r^2−x^2).

EDIT: I mean honestly none of these exercises were taught in the videos at all
• The input that will maximize a positive expression will also maximize its square, as the squaring function is a strictly increasing function (on the domain of non-negative real numbers). Sometimes you have to think a bit out of the box to apply these tools to certain problems.
• for #1, how should I explain how or why at x=-1 and at x=3 they're absolute max and mins.

Same for #2: For the inflection point at x=1, should I explain by saying that the slope changes signs?
• Since you had a defined, closed interval, the largest value is the absolute max FOR THAT INTERVAL. Similar for the min.
You can show that x=-1 is an absolute max because it is the only local max, the function is continuous, and there are no other extrema on the given interval. Thus, it must be the largest value of f(x) over the interval. In other words, since f(x) is increasing over -3<x<-1, then f(-1) must be the largest value of f(x) over -3<x<-1; and, since f(x) is decreasing over -1<x<3, then f(-1) must be the largest value of f(x) over that interval. Thus, f(-1) must be the absolute max over the given interval.

You can show that f(3) is the absolute min on the given interval because the function is continuous, reaches a max along the interval, has no other local extrema, and f(x) is decreasing as x increases for the interval -1 < x ≤ 3. Thus, the absolute minimum over the given interval must be either f(-3) or f(3), and f(3) is less than f(-3).

You should prove that f(1) is an inflection point with the Second Derivative test:
f''(1) = 0, a necessary condition for an inflection point.
f''(1-ε) is positive AND f''(1+ε) is negative, a sufficient condition for an inflection point. (Note: ε means an arbitrarily small number).
Therefore, f(1) fully meets all the requirements of the Second Derivative Test, thus it must be an inflection point.
• At , Sal finalizes our answer as a number with a decimal in the numerator and fractions in the denominator.

To express our answer in simplest terms, do we need to combine the two fractions in the denominator?

Also, is it the custom in mathematics when simplifying fractions to give our final answer in either decimals or fractions, but not both, or is it "permissible" to leave our answer in its current form.
• Since Sal's goal is to find a decimal approximation for x, it is fine to leave his work in the form it's in at . If this was to be his final answer, then yes it is conventional to rationalize denominators, not leave fractions with decimal parts, and simplify fractions using a common denominator. The advantage of leaving it the way he does is that the 'fraction' he creates is still an exact value for x. Since he uses a calculator to compute the final answer, no 'clean-up' was necessary.
(1 vote)
• At Sal differentiates (100-x)^2/16 as (100-x)/8(-1), wouldnt you just use the chainrule or even the quotient rule? I thought the derivative would've been ((2(100-x)(16)-(100-x)^2))/(16^2) ?
(1 vote)
• At , why is the derivative of sqrt(3)x^2/36 = root 3x/18? I tried using chain rule and quotient rule on that term and I ended up getting 4 root 3x as my final answer. Was there an easier way to do that, or a correct way rather?
• No need for the chain rule or the quotient rule here. All you have is x^2 times a constant (which happens to be sqrt(3)/36). The derivative of x^2 is 2x, and you multiply that by the constant to get the result indicated.
• Am I right in thinking that the use of the 2nd derivative test is not necessary in order to verify a min/max value at a critical point?

Using the example in the video - we know that the CP = 0 at x = 56.5. I know this final calculation is done at the end but you can easily do it when you are establishing the CP=0 value.

Then by taking 2 values- one slightly above and one below (eg 55.5 and 57.5) and plugging them back into the 1st derivative equation you get -0.22 and +0.22 - meaning the curve is transitioning from negative to zero to positive.

This confirms an upward curve and thus x =56.5 as the minimum value.

Am I correct in my thinking?