[Pri20150220FSA] A Fishy Shaped Area

Question

Plan of Attack

     This problem looks difficult because the shaded area does not seem to look like any regular shape.  Is it a fish whose head is pointing in the top left direction and whose tail is in the bottom left direction?  Fortunately, this is not a Rorschach ink-blot test.

     As with all “area” problems in primary (elementary) school, we try to break down the unfamiliar shape into regular shapes (e.g. parts of circles, squares, triangles, rectangles).  It is basically a divide-and-conquer strategy (using heuristics H10 & H11).  If we look carefully, we realise that the required area consists of a semi-circle less a funny horn-shaped area, which I call ‘F’.  F is for funny, for want of a better description.  So I am going to find the area of the semicircle (which is half of a circle), then subtract the area of F.  We’ll worry about finding the area of F later.

Solution

     Area of semi-circle [in cm²]

= ½ ´ p ´ radius²

= ½ ´ p ´ (5) ² = ²⁵/₂ p   

     It is good to leave the calculation with  p  until the last step.

     OK, we are done with the first part.  [Heuristic H11]  Let us us tackle the next part, which is to find the area of  F.  Note that this is a 45°-45°-90° isosceles triangle minus a 45°-degree sector (which is one-eighth of a circle, because ^45°/_360° = ¹/₈).

     Area of F  [in cm²]

= Area of triangle – area of sector

= ½ ´ 10 ´ 10 – ¹/₈ ´ p ´ radius²

= ½ ´ 10 ´ 10 – ¹/₈ ´ p ´ (10) ²

= 50 – ²⁵/₂ p

     Let us combine our answers.  We need to subtract 50 – ²⁵/₂ p  from the area of the semi-circle.  If we subtracted 50 from ²⁵/₂ p, we would have over-subtracted.  So we need to add back ²⁵/₂ p.  Hence

    Required Area [in cm²]

= Area of semi-circle – area of F

= ²⁵/₂ p  – (50 – ²⁵/₂ p)

= ²⁵/₂ p  – 50 + ²⁵/₂ p

= 25p  – 50

Using the calculator’s value of p,  we obtain

     Required Area = 28.54 cm²  (to 2 decimal places)

Heuristics Used

H10. Simplify the problem

H11. Solve part of the problem

Commentary

     This difficult problem was solved by dividing the problem into smaller pieces and tackling each piece one at a time.  We break down a complicated shape into familiar shapes.  That is the secret.

Please refer to this similar problem.