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Ladder and Cube

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

Growing

Which is larger: (a) 1.000001^{1000000} or 2? (b) 100^{300} or 300! (i.e.factorial 300)

Archimedes and Numerical Roots

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

Back Fitter

Age 14 to 18
Challenge Level

After a while spent as a toughnut, we recieved the solution to this problem. We were very pleased to see that one of our younger solvers, Jonathan, realised that the first graph was $y=0.5 x$. Impressively, a full solution was sent in by James from Bay House, where all of his functions gave a close fit with the data -- well done James!

James' suggestions agreed with ours in six of the cases (bold font), but differed in four cases (normal font). Perhaps you might like to consider which you feel are the closer fit?

Experiment 1: $y=x/2$

Experiment 2: $y=\sin(x) $

Experiment 3: $y=x^2$

Experiment 4: $y=x-\sin(2x)$

Experiment 5: $y=5\log_{45}(x+1)$ $\left(\mbox{we got }y=\sqrt{x}\right)$

Experiment 6: $y=(\sin(1.7x)+1)/2$ $\left(\mbox{we got }y=\sin^2(x)\right)$

Experiment 7: $y=-0.6+(\log_{10}(6x))^{-1}$ $\left(\mbox{we got }y=\frac{1}{1+x^2}\right)$

Experiment 8: $y=\log_{15}(7x+1)$ $\left(\mbox{we got }y=1.65x/(1+x)\right)$

Experiment 9: $y=\cos(2x)$

Experiment 10: $y=2^x$