# A maximizing area question

April 6, 2018 pm30 12:30 pm

I gave this question to students as a challenge at the end of a trig unit.

A quadrilateral has perimeter = 60 and a 30º angle. What is the maximum possible area?

I think this is cute. The kids had to make some assumptions, test them, and use trigonometry along the way. It’s not “open-ended” but it does involve some investigation, and it is not just a direct application of what I’ve taught them.

What’s your answer?

Do you like the question?

And do you know why “What is the minimum possible area?” is not a good question?

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7 Comments
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Had to make a GeoGebra sketch for looking into this. Trying to figure out how to pitch it for my calculus students. https://www.geogebra.org/m/ZDJbE4tK

I did this with students without calculus, we made arguments that appealed to the maximum value of sine, maximum altitudes, and to symmetry.

As a calculus problem it’s very challenging! Although I also feel like the symmetry arguments are tricky, because of that fixed angle. (The kind of sliding around of things I want to do tends to disrupt angles.)

Try setting AB to 20, and AD to 0.001.

It works with a rhombus, which is a skewed square. A square maximizes area. Therefore each side should be equal to 15 units where the angles are 30, 30, 150 and 150. Using the trig formula of A = (s)(s) sine (angle), you get a maximum area of 112.5 square units. You cannot minimize area because of the constraints of the angle and perimeter. Whadda think?

I posted this question on MathStackExchange [https://math.stackexchange.com/q/2760727] where it was pointed out that there should be an assumption around convexity: otherwise, you can create a concave figure that approaches an equilateral triangle to maximize the area. See the link above and its answers – some of which contain some tough mathematics! – for more information [including some generalizations].

Thank you for extending the discussion, and exhausting the issues with the problem statement. The stackexchange discussion was interesting for me.

I’ve often wondered about how much information uniquely describes quadrilaterals…