Problem $7$: $[15]$ An empirical formula used for determining the natural frequency associated with translational motions of a steel chimney has the form

${}_n=7100\frac{(d)}{h^2}ra\frac{d}{s}$

where $d$ is the diameter of the chimney and $h$ is the height of the chimney. The damping factor associated with a chimney of diameter $7$ m and height of $100$ m is $0\mathrm{.}002.$ This chimney is located in a region where wind gusts are capable of producing initial displacements of $0\mathrm{.}1$ m and initial velocities of $0\mathrm{.}4\frac{m}{s}.$ Determine the maximum displacement experienced by the chimney and check if it satisfies the construction codes, which require the maximum displacement to be less than $4\%$ of the diameter. If the present chimney does not satisfy this code, what design changes would you propose so that the construction code is satisfied?