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Year 10 Interactive Maths - Second Edition


Problem Solving Unit

Problem 14.1  A Ladder



Problem 14.2  A Speedboat

Five equally spaced buoys represent the direct passage across Lake Theta from the Stanford to the Fairview landing.

Outboard and Speedboat depart from Stanford in their respective boats to go to Fairview.
Outboard advances direct whereas Speedboat needs to stop at the Cranfield landing to refuel.

Outboard just passes the second buoy as Speedboat stops at Cranfield. Outboard is passing the third buoy as Speedboat leaves Cranfield.

They landed at Fairview simultaneously.

a.  How far is Cranfield from Stanford?
b.  How far is Fairview from Cranfield?


Problem 14.3  The Crease

How long is the crease that is made by folding a 15 cm by 8 cm piece of paper so that two opposite corners of the papers coincide?


Problem 14.4  A Prism

A rectangular prism's surface area is equal to 324 cm2.  If its length and width are 12 cm and 8 cm respectively, find the prism's height.


Problem 14.5  Christmas Cards

Determine the maximum number of 20 cm by 15 cm Christmas cards you can make from a 3 m by 2 m cardboard sheet, if each card has four pages (i.e. 2 leaves).


Problem 14.6  A Car's Engine

A car's engine has 6 cylinders, each of diameter 9.6 cm and height 10.4 cm. Find the capacity of the engine's cylinders in cc (cubic centimetres).


Problem 14.7  A Closed Cube

A cube's volume is 125 cm3. Find the cube's side-length and hence its surface area.


Problem 14.8  Capacity of an Engine

A car's engine has 4 cylinders, with a total capacity of 2000 cc.
a.  If the diameter of each cylinder is 9 cm, calculate the height of each cylinder.
b.  If the height of each cylinder is 8 cm, calculate the diameter of each cylinder.


Problem 14.9  Capacity of a Pool

A rectangular swimming pool's depth is 1 m at the shallow end and gradually increases to 4 m at the deep end, as shown in the following diagram.

Calculate:
a.  the cost of painting the pool at $5 per square metre
b.  the amount of water (measured in litres) required to fill the pool


Problem 14.10  Capacity of a Pool

A swimming pool has a length of 50 m and a width of 36 m.  The pool's depth is 1 m at the shallow end and gradually increases to 2.5 m at the deep end. Calculate the capacity of the pool in litres.


Problem 14.11  A Steel Pipe

The cross-section of a 12 m long steel pipe is an annulus with inner and outer radius of 14 cm and 21 cm respectively.  Calculate the volume of metal used to make the pipe.


Problem 14.12  A Square Sheet

A square sheet of metal 24 cm by 24 cm has a square piece cut out from each corner.  The remaining piece is folded to form a cube with an open top.

Develop a formula that can be used to calculate the volume of the cube.


Problem 14.13  Water Tank

Half of a 7 m by 6 m by 2 m rectangular prism is filled with water.  The water is allowed to flow out at the base of the prism into a cylindrical tank of diameter 3.2 m.  Calculate the height to which the water will rise in the tank.


Problem 14.14  Slicing a Cube

1.  Can you obtain a cross-section of an equilateral triangle by slicing a cube?  Justify your answer by stating the reasons.

2.  Can you obtain a cross-section of a regular hexagon by slicing a cube?  Justify your answer by stating the reasons.

3.  Can you obtain a cross-section of a regular pentagon by slicing a cube?  Justify your answer by stating the reasons.


Problem 14.15  Two Spheres


Problem 14.16  Shoemaker's Knife

A shoemaker's knife consists of a large semicircle and two small semicircles with diameters that add up to that of the large semicircle, as shown in the following diagram.

1.  Use a compass to draw a shoemaker's knife.

3.  Find the area of the face of the knife.

4.  If the thickness of the knife is 1 mm, find the volume of the knife.


Extension

5.  Draw a shoemaker's knife on graph paper.

6.  Draw a line PQ perpendicular to AB; and draw a circle of diameter PQ as shown in the following diagram.

Your report should include all working and diagrams.


Problem 14.17  Paper Rolls

Paper is wrapped on a roll with an outer radius of 14.61 cm and an inner radius of 4.45 cm, and a paper thickness of 0.008 cm. The inner radius forms a cylindrical core around which the paper is wrapped.

a. Find N, the number of wraps around the core.
b. Find L, the length of the paper wrapped on a roll.
c. If the paper is being unwrapped at a constant rate of 3 metres per minute, find the time, T, remaining until the paper supply is exhausted.


Extension

Assume that the paper is stored on a roll with an outer radius of R cm and an inner radius of r cm and paper thickness of t cm. The inner radius forms a cylindrical core around which the paper is wrapped.

d.  Find N (number of wraps) in terms of R, r and t.
e.  Find L (paper length) in terms of r, t and N.


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