29th October 2004 at 01:00
Use all the numbers: 1, 2, 3, 4, 6 and 8.

(a) Put one number in each circle so that the product of the three numbers on each side of the triangle is 24.

(b) Find another way of putting the numbers in the circles so that the product of the three numbers on each side of the triangle is the same.

P is a point on a sphere of radius 6cm.

A pair of compasses is opened to make a radius of 4cm and a circle with centre P is drawn on the sphere.

(a) What is the radius of the circle?

(b) To what radius must the compasses be opened to draw the largest possible circle with centre P?

SOLUTIONS Equal products Product: 24 Product: 48 Circles on spheres (a) The radius of the circle is 8C23 cm. Using Pythagoras' theorem: r2 = 62 - (6 - a)2 = 42 - a2, giving a = 43 and r = 8C23. (b) The largest circle that can be drawn is round the "equator" of the sphere, when c, the radius of the compasses, is given by: c2 = 62 + 62, or c= 6C2.

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