SEMI-CIRCLE
To find the area of the semi-circle we use the equation ½∏r2 (where ∏ equals 3.14).
I cannot work out the area of the semi-circle as I do not know the value of R.
CIRCUMFERENCE OF FULL CIRCLE = 2 ∏r
CIRCUMFERENCE OF HALF CIRCLE = ∏r
50 = ∏r
50
= r
15.92 = r (2d.p)
I can conclude from my findings that the semi-circle gives an overall area of:
AREA = 392.5 cm
VOLUME = area x 200cm
VOLUME = 78500
RECTANGLE
If I have the two sides equal and the base one length then hopefully I can create the largest area. To find the best area I will have to find the best length for each side.
A+B+C = 50cms
The formula I will use to find the area of the rectangle is:
BASE x HEIGHT
To find out which would be the best measurements to use I am going to increase the height of each side by one until I get to 24. I will stop at 24 because this is the most I can increase the height by and make sure that when the base and height is added it will equal 50. I will find the area for each one that I go up by.
*BOLD indicates turning point.
Triangle
A = 25cm
B = 25cm
I am now going to investigate the triangle so I can find the largest possible area. I know that A and B are equal because there is a 50cm strip which is halved and equals 25cm per side.
The possibilities for the size of the angle between sides A and B are from 1 to179. But if I use the sin rule, which I know I can use because I know the angle of the two sides, I can work out the area for the triangle easily.
SIN RULE
½ab Sin C
50
½ x 25 x 25 x 50 Sin
239.389cm
I will now proceed to carry out the calculations for the triangle and I will display my findings in the table below.
*BOLD indicates turning point.
Trapezium
By looking at the trapezium I can see that it is made up of two triangles and a rectangle.
We found B as 25cm because when we looked at the findings from the rectangle 25cm gave the best area for our shape.
To find the area for the trapezium I am not going to change the lengths of the side instead I will change the angle θ. The values for the angles can range from 1 to 89. For calculating each of my angle sizes I will go up in steps of 5.
To find the area of the trapezium I will use the formula:
Area of trapezium = ½ the sum of the parallel sides by the
Perpendicular height.
Working Out For Trapezium
To find X we use the equation:
Sin45 = X/12.5
X = 12.5 Sin45
X = 8.8388cms
To find Y we use this equation:
Y = 2X+25
= 2(8.8388)+25
= 42.6776
To find H we use the equation:
Cos45 = H/12.5
H = 12.5 Cos45
H = 8.8388cms
Check H using Pythagoras’ Theorem
H2 + 8.83882 = 12.52
H2 = 12.52 - 8.83882
H = 8.8388cms
We can now apply our three unknowns into the final formula:
I now know that my final area for the trapezium is; 299.09cm2
Test Tube
The test-tube is the most complex shape here because it consists of a rectangle and a semi-circle. Because of the shape consisting of a rectangle and a semi-circle there will be a lot more working out.
I am going to calculate the area and increase the height of the rectangle each time by 1cm.
Calculating The Area
I know that the area of a semi-circle = ½ ∏r2 (∏ = 3.14)
We cannot use this because we do not know the circumference, however we can calculate the circumference by using the equation:
Πr
48 = Πr
48/3.14 = r
- = r
We can now work out the area of the semi-circle:
= ½ Πr
= ½ x3.14x15.2872
= 366.97
To find the area of the rectangle we use the equation:
= h x b
Base = 1 x r
Base = 1 x 15.287
Diameter = 15.287
Area = 1 x 15.287
= 15.287
We can now finally work out the area of the test tube:
Area of test-tube = Area of Semi-circle + area of rectangle
= 366.897 + 15.287
= 382.184
Conclusion
Now that I have investigated all five shapes I can now display my findings in order in the table below. The shapes are listed from the one with the largest area/volume to the one with the smallest. The one with the largest area/volume is the best shape for a drain.
I can now see that the best drain is the semi-circle as it has the greatest area and volume.
Extension
If I had more time I would have been able to investigate after shapes and went on to investigate about different sizes of plastic and how they effect the area and volume of the drain.
Evaluation
I found this project very rewarding and I thoroughly enjoyed it. I was pleased when at the end I found that the semi-circle was the shape that gave the greatest area and volume.
Although I enjoyed most of the project I found some parts quite difficult, for example the area of the trapezium had a lot of complex working out to do.
There are parts of the project that I could have made more accurate by rounding to a higher decimal place or using the Π button on the calculator.
