Sunday Afternoon Maths XXX

The coins are 50¢, 25¢, 10¢, 10¢, 10¢ and 10¢

What is the maximum possible amount of money which could be in the till so that no change could be made for any of the coins?

Let \(t\), \(u\) and \(v\) be the prortion of a hole dug in one day by Timothy, Urban and Vincent respectively. Then the following equations hold:

These can be solved to find that \(t=\frac{1}{24}\) so it will take Timothy 24 days to dig a hole.

Working alone, how long does it take Urban to dig one hole?

Working alone, how long does it take Vincent to dig one hole?

Call the length of a side of the square \(x\). CAB and CEF are similar triangles, therefore:

Consider a tetrahedron with vertices at \((0,0,0)\), \((a,0,0)\), \((0,b,0)\) and \((0,0,c)\). A cube is placed in the tetrahedron so that three of its faces are in the xy, xz and yz planes and one of its vertices lies on the other face.

What is the length of a side of the cube?