# Geometry/Geometry

Question
a)Figure A shows a water trough of length 14cm. The cross section is in the shape of an isosceles trapezium. The lengths of the parallel sides of the trapezium are 20cm and 16cm respectively.
Find the volume of water needed to fill the trough to a height of 5cm.

b)Figure B is a solid metal rod made up of a cone with height h, a cylinder and a hemisphere of radius r. The length of the cylinder is 4 times the height of the cone. The volume of the cone and the volume of the hemisphere are the same.
Show that h=6cm.

a)Since it is 40 cm long and 20 cm wide, that is 40x20 = 800 cm².
Since the depth is 5 cm, take 5x800 to get the volume of cm³.

b) The length of the cylinder is 4h.

The volume of the cone is pi*r²h/3.
The volume of the hemisphere is 4*pi*r³/6.

These are equal, giving pi*r²h/3 = 4*pi*r³/6.
Multiplying by 6/(pi*r²) gives 2h = 4r, so r = 2h/4 = h/2.

That makes the total length of the piece h + 4h + h/2 = 11h/2 = 11r.
Since we have 33 = 11r, r = 3.  Since 2r = h, h is 6.
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#### Scott A Wilson

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