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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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4 Comments leave one →
  1. April 7, 2018 am30 11:52 am 11:52 am

    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

    • April 7, 2018 pm30 6:43 pm 6:43 pm

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

      • April 10, 2018 am30 9:46 am 9:46 am

        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.)

  2. Diane Mammolito permalink
    April 12, 2018 am30 8:14 am 8:14 am

    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?

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