HSC General Mathematics

1. HSC General Mathematics M5: Applications of area and volume

2. What is part of M5 measurement? Areas of ellipses, annuluses and parts of a circle Errors in calculations Calculating areas of composite figures M5: Further applications of area and volume Volume of composite solids Surface area of spheres Applying Simpson’s Rule Surface area of Cylinders

3. Formulas that are not on sheet • Pythagoras theorem • Circumference of circle • Area of circle • Area of triangle • Area of rectangle • Area of parallelogram • Area of trapezium • Area of rhombus • Volume of Prism

4. Answers to formulae that are not on sheet • Pythagoras c²=a²+b² (The square on the hypotenuse is equal to the sum of the squares on the other two sides.) • Circumference of a circle C=2Πr • Area of a circle A = Πr² • Area of a triangle A = ½bh OR A =bh/2 • Area of a rectangle A =bh • Area of a parallelogram A = bh • Area of a trapezium A=h/2 (a+b) • Area of a rhombus A=Dd/2 • Volume of a prism V=Ah (Area of the cross-section x height)

5. M1 and M2 – What was involved? • Determining appropriate units to use • Conversion between commonly used units • Accuracy in measurement • Error in measurement • Significant figures, scientific notation • Rates and ratios • Area of triangles and quadrilaterals • Field diagrams • Classifying polyhedra • Surface area • Volume and capacity

6. Circlework • Area of annulus =area of big circle – area of small circle = Π(R² – r²) • Area of an ellipse = Πab • Where a=length of semi-major axis • And b=length of semi-minor axis • Area of a sector = Θ/360 x Πr² • Arc length ℓ= Θ/360 x 2Πr

7. Composite figures • Remember that more than one method may be used, adding or subtracting are both acceptable.

8. Simpsons rule, one application • Simpson’s rule is used to find the area of an uneven field where one side is a curved boundary. • A=h/3(d₁ + 4.d₂ + d₃) where h= width of strip (between successive measurements) d₁ = first distance d₂ = middle distance d ₃ = last distance

9. Simpson’s Rule – Multiple applications • You can either do two or more separate applications or you can put the first and last in brackets and then 4 times the even slotted distances and 2 times the odd slotted distances.

10. Surface areas of cylinders • Read the question carefully to determine whether it is open or closed and whether it is open both top and bottom • If open and asking for the surface area of the curved surface only, then • SA = 2Πrh (if you cut longways through the cylinder you would have a rectangle with the breadth being the circumference of the circle , thus 2Πr, and the height of the cylinder being h.) • If a closed cylinder then you have to find the area of the circular base and add that in. • i.e. A closed cylinder with top and bottom is • SA = 2Πrh + 2 x Πr²

11. Surface area and volume of a sphere • Surface area of a sphere • SA = 4Πr² • Volume of a sphere • V = 4/3Πr³

12. Volume of composite solids • Volume of composite figures can be found by adding the volume of multiple different solids or subtracting the volume of one solid from another if looking for remaining spaces.

13. Determining error • The accuracy of measurement is ± the smallest unit of the measuring instrument. • (If it is a ruler measuring in cm’s then the error is 0.5cm or 5 mm. If it is a set of scales measuring in 100gram increments then the error is ±50 gram.) • Percentage error • % error = (difference ÷ original) x 100% • Or • % error = (½ the smallest unit ÷ actual measurement) x 100%

14. Hsc questions and what gets best marks • When answering questions do not round off during a calculation. Continue to use full calculator display and write down this as your solution before writing a concluding statement with a rounded answer. Rounding off too early causes significant differences in the final result. You can obtain marks for correct rounding even if your answer is incorrect.

15. Past Questions • On attached sheets