Measurement and Trigonometry - October 2004, Stage 4&5

Problems

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Square It

Stage: 1, 2, 3 and 4 Challenge Level: Challenge Level:1

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

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Salinon

Stage: 4 Challenge Level: Challenge Level:1

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

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Sliced

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

An irregular tetrahedron has two opposite sides the same length a and the line joining their midpoints is perpendicular to these two edges and is of length b. What is the volume of the tetrahedron?

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Tubular Stand

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

If the radius of the tubing used to make this stand is r cm, what is the volume of tubing used?

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Bend

Stage: 5 Challenge Level: Challenge Level:1

What is the longest stick that can be carried horizontally along a narrow corridor and around a right-angled bend?

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Complex Sine

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Solve the equation sin z = 2 for complex z. You only need the formula you are given for sin z in terms of the exponential function, and to solve a quadratic equation and use the logarithmic function.