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Moss's Egg. Task -1- Find the area of the shaded region inside the two circles shown below. The two large circles have a radius of 6cm.

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Introduction

Moss’s Egg

The following formulas were used in the solving of the questions of this assessment piece:

Area of a Circle        

image06.png

Circumference of a Circle

image07.png

Area of a Sector

image43.png

Arc Length

image46.png

Working Out and Explanation

Task -1- Find the area of the shaded region inside the two circles shown below. The two large circles have a radius of 6cm. Their centres are A and B.

image00.png

image01.png

From the information given above, we know that the radii of the two larger circles are 6 cm in length. We define the radius of the circle as a straight line extending from the centre of a circle to its circumference. Since we know that points A and B are the centres of the two large circles, we can conclude that this is the length of the two points from A to B is 6 cm also, since point B is along the circumference of the top larger circle, and vice versa. From lengths A to B is the therefore the diameter of the smaller circle between the two larger ones, and thus we can conclude that the radius of the smaller circle is 3 cm. The area of the small circle can therefore be calculated using the formula indicated:                  image06.pngimage06.png, where A equals the area and r is the radius. Thus: A = image27.pngimage27.png (32) = 9 image62.pngimage62.png

 28.3 cm2

image00.png

Task -2- The same circles are shown below.

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Middle

image52.png= image53.pngimage53.png= image54.pngimage54.png= 450

Thus angle ACE is 900 and therefore is right angled. As ACB and DCE are vertically opposing angles, ∠ DCE is also 900. We can now substitute the information we have to determine the area of the sector: image50.pngimage50.png where image53.pngimage53.png is 900 and r is 6- image45.pngimage45.png.

image55.pngimage55.png22.43cm2

Task -4- Find the area of the sector BAE in the diagram below.

To find sector BAE in the circle, we also must apply the formula: image56.pngimage56.png . In the diagram displayed below, two internal angles of the two right angled triangles have been found. Therefore, the quickest method to determine ∠BAE is to subtract 1800 from the two known angles, as it is known that all internal angles of any triangle is equal to 1800.

∴ ∠BAE = 180 – 90 – 45

                = 45 0

The second value needed in calculating this sector is the radius. ∠BAE falls into both larger circles of the diagram. The length AB is the radius of these two circles, which we know to be 6cm in length. Therefore, the radius for determining the area of this sector is 6 cm. Thus, we simply substitute into the formula. image04.png

image57.pngimage57.png2

                ≈ 14.14 cm2

Task -5- Look for the shape of an egg in the enclosed areas of the circles shown below. This is called Moss’s Egg. Find the area and perimeter of Moss’s Egg.

image04.png

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Conclusion

>

Sector DCE = image12.pngimage12.pngimage11.pngimage11.png = image13.pngimage13.pngr

Thus, in order to find a formula for determining the area of Moss’s Egg, the following equation must be used, with substituted figures in terms of r.

A Moss’s Egg = image14.pngimage14.png A small  AB + A sector BAE+ A sector ABD + A sector DCE – ΔABC

 A Moss’s Egg   = image15.pngimage15.png2 + image16.pngimage16.pngr2 + image16.pngimage16.pngr2 +                   image17.pngimage17.png2image18.pngimage18.pngimage19.pngimage19.png

= image20.pngimage20.png2 +image21.pngimage21.png2 + image22.pngimage22.pngimage23.pngimage23.png2 - image24.pngimage24.png2

= r2 (image25.pngimage25.png +image26.pngimage26.png + image22.pngimage22.pngimage27.pngimage27.png-image28.pngimage28.png)

= r2 (image29.pngimage29.pngimage30.pngimage30.pngimage31.pngimage31.png)

                ≈ 0.995r2

To test this formula, the radius 6cm can be substituted into r, which allows the answer calculated by this formula to be compared with that of question 5a.

0.995(6)2 = 35.82 cm

Therefore, the suggested formula has been supported.

  1. Much like section A of this question, finding the perimeter of the egg consists of substituting the calculated figures in the previous questions into their relationships in terms of r. Therefore, a formula for the perimeter of Moss’s Egg can be calculated using the method displayed below.

C semicircle + Arc AD + Arc BE + Arc DE

                = image32.pngimage32.pngimage33.pngimage33.png2image34.pngimage34.png + image35.pngimage35.png + image37.pngimage37.png(image38.pngimage38.png)

                = image39.pngimage39.png x image40.pngimage40.png

                = image41.pngimage41.png                                = image42.pngimage42.png

                ≈ 3.6r

To test this formula, substitute r to 6 and compare with answer calculated from question 5b.

3.6 x 6 = 21.6 cm

Therefore the formula above has been supported.

Bibliography

Haese, S et al. 2006, Mathematics For The International Student. Haese & Harris         Publications, Australia


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