A person has a weight of 500 N. If he wants to lie on a bed of nails, the pressure must be less than 5N/cm2. Each nail has an area of 0.005cm2. How...

By definition, pressure is the force over the area on which it is uniformly distributed, and in our case, the force considered to be perpendicular to that area as well. 

Denote the number of nails as K, then their total area A will be `0.005*K cm^2.` The force F is known: it is 500 N. So the pressure `P = F / A = 500 / (0.005 * K) = 10^5 / K (N / (cm^2)).`

And by the conditions it must be less than `5 N / (cm^2).` This is an inequality and we can solve it for K.

`10^5 / K lt 5` is equivalent to `K gt 10^5 / 5 = 20,000.` This is the answer: the number of nails must be greater than 20,000.