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Perimeter & Circumference

Directions: Using the digits 1-6, at most one time each, fill in the boxes to create the largest and smallest combined perimeter/circumference for the rectangle and circle.

Hint

Hint

Which box number change has the greatest effect on the combined perimeter/circumference? Why?
What part of your calculations will be the same every time?

How many permutations are there of the numbers 1-6 in four positions?
Which permutations matter for the largest sum? Why?
Which permutations matter for the smallest sum? Why?

Answer

Answer

greatest: (53 · 6) + (2π · 4) = 343.13… units

least: (24 · 1) + (2π · 3) = 42.85… units

Source: Christin Smith

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16 comments

  1. The solution given is actually for area of the rectangle and circumference of the circle

    • Where do you put the numbers? Like which boxes? Thanks.

      • You can choose to use any of the digits 1-6 in each of the boxes so that the rectangle has side lengths and the circle has a radius. You can only use each digit once.

    • David, you are right. With the solution (with given numbers) should look like:
      greatest: (2*53+2*6)+(2π · 4)=143.13….
      least: (2*24+2*1)+(2π · 3)=68.84…..

      HOWEVER: Neither are the greatest or smalles that we found.
      Greatest: (2*65+2*4)+(2π · 3)=156.84…
      Least: (2*13+2*4)+(2π · 2)=46.56…

  2. Greatest: Rectangle with sides 63 and 4 and circle with radius 5. Answer: 165.4

  3. Least: Rectangle with sides 13 and 4 and circle with radius 2. Answer: 46.56

  4. Are we finding the smallest perimeter or area???

  5. My students came up with the following for greatest:

    2(64+3) + (5*2*pi) = 134 + 31.4 = 165.4

  6. Richard Buettner

    I wrote a program that solves these. The largest combined perimeter/circumference is the one that was already found with:

    A 64 x 3 rectangle and a circle of radius 5 = 165.42
    A 63 x 4 rectangle and a circle of radius 5 = 165.42

    Least:

    A 14 x 3 rectangle and a circle of radius 2 = 46.57
    A 13 x 4 rectangle and a circle of radius 2 = 46.57

    • Richard Buettner

      For combined areas:

      Greatest:
      A 53 x 6 rectangle and a circle of radius 4 = 368.27

      Least:
      A 34 x 1 rectangle and a circle of radius 2 = 46.57

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