HSC General Mathematics

M5: Applications of area and volume

What is part of M5 measurement?

M5: Further applications of area and volume

Areas of ellipses, annuluses and parts of a circle

Calculating areas of composite figures

Applying Simpson’s Rule Surface area of

Cylinders

Surface area of spheres

Volume of composite solids

Errors in calculations

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

Formulas 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)

Answers to formulae that are not on sheet

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

M1 and M2 – What was involved?

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

Circlework

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

Composite figures

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 distanced₂ = middle distanced ₃ = last distance

Simpsons rule, one application

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.

Simpson’s Rule – Multiple applications

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

Surface areas of cylinders

Surface area of a sphere SA = 4Πr²

Volume of a sphere V = 4/3Πr³

Surface area and volume of a sphere

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.

Volume of composite solids

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%

Determining error

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.

Hsc questions and what gets best marks

On attached sheets

Past Questions