The Volume of a pyramid with a square base is 200 cubic cm. The height of the pyramid is 13cm. What will be the length of the slant edges (i.e. the distance between the apex and any other vertex), rounded to the nearest integer?

Volume of the pyramid = $$200$$ cubic cm. 
The volume of  a pyramid is usually a third of the volume of a cuboid of the same height. 
Therefore, a square cuboid of the height of the pyramid will have a volume of $$600$$ cubic cm.

We know that the height is $$13$$ cm.
Area of the base square* height = $$600$$ cm.
=> Area of the base square = $$\frac{600}{13}$$ cm$$^2$$.
Side of the base square = $$\sqrt{\frac{600}{13}}$$ cm.

Length of diagonal of the base square = $$\sqrt{\frac{600}{13}}*\sqrt{2}$$

Now, the height of slant edge of the pyramid can be found out by using the Pythagoras theorem. 
Length of half the diagonal of the base square will form one of the sides and the height of the pyramid will form the other side. The slant height of the pyramid will be the hypotenuse of the right-angled triangle. 

Height of slant edge = $$\sqrt{13^2 + \frac{1}{4}*2*\frac{600}{13}}$$
                   = $$\sqrt{169 + \frac{600}{26}}$$
                   = $$\sqrt{\frac{4394+600}{26}}$$
                   = $$\sqrt{ \frac{4994}{26}}$$
                   =$$\sqrt{192.07}$$
                   =$$13.85$$ cm

The nearest integer is $$14$$. Therefore, option C is the right answer.