Alan

Here's my take on this:

You use the fact that the sine of an angle is opposite over hypotenuse here.

opposite = 618 - 9 = 609m

so sin(76°) = 609/h

h = 609/sin(76°)

$${\mathtt{h}} = {\frac{{\mathtt{609}}}{\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{sin}}{\left({\mathtt{76}}^\circ\right)}}} \Rightarrow {\mathtt{h}} = {\mathtt{627.643\: \!700\: \!274\: \!085\: \!654\: \!1}}$$

so h = 627.64m to the nearest hundredth of a meter.

Or you could lay it out like this:

There is no regular polygon with diagonals 35 and apothem 10.  The radius would be 17.5 so the cosine of the angle between an apothem and the nearest radius would be 10/17.5 this means the angle would be

cos^-1(10/17.5), so the angle between radii would be 2*cos^-1(10/17.5).

$${\mathtt{2}}{\mathtt{\,\times\,}}\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{cos}}^{\!\!\mathtt{-1}}{\left({\frac{{\mathtt{10}}}{{\mathtt{17.5}}}}\right)} = {\mathtt{110.300\: \!190\: \!841\: \!908}}$$

360° divided by this doesn't result in an integer!

Using the calculator here:

$${\mathtt{sinX}} = {\left({\mathtt{4.5}}{\mathtt{\,-\,}}\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{sin}}{\left({\mathtt{45}}^\circ\right)}{\mathtt{\,-\,}}\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{tan}}{\left({\mathtt{75}}^\circ\right)}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)} \Rightarrow {\mathtt{sinX}} = {\mathtt{0.393\: \!310\: \!438\: \!163\: \!336\: \!6}}$$

$${\mathtt{X}} = \underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{sin}}^{\!\!\mathtt{-1}}{\left({\mathtt{0.393\: \!310\: \!438\: \!163\: \!336\: \!6}}\right)} \Rightarrow {\mathtt{X}} = {\mathtt{23.160\: \!642\: \!003\: \!775^{\circ}}}$$

X ≈ 23.16°

Could also have X = 180°-23.16° as sin is positive in the second quadrant.

Consider the octagon to be made up of 8 triangles each of which look like the image below:

Angle β is (180-45)/2 = 67.5°

Use the sin rule to get s

s/sin(45°) = 1.4/sin(67.5°)

$${\mathtt{s}} = {\frac{{\mathtt{1.4}}{\mathtt{\,\times\,}}\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{sin}}{\left({\mathtt{45}}^\circ\right)}}{\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{sin}}{\left({\mathtt{67.5}}^\circ\right)}}} \Rightarrow {\mathtt{s}} = {\mathtt{1.071\: \!513\: \!610\: \!623\: \!269\: \!6}}$$  

s ≈ 1.07m

15% means 15/100, so 15% of 334 is (15/100)*334

$$\left({\frac{{\mathtt{15}}}{{\mathtt{100}}}}\right){\mathtt{\,\times\,}}{\mathtt{334}} = {\frac{{\mathtt{501}}}{{\mathtt{10}}}} = {\mathtt{50.1}}$$

The Anonymous answer is almost correct, except that he/she should have added 13 rather than subtracting it.

3⁵⁰⁰ ≈ 3.636×10²³⁸

For those who don't understand why heureka manipulated reciprocal times (which is the correct thing to do) here's a more long-winded approach.