Radius of maximum profit for a spherical building

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A building shaped like a sphere of radius $r$ is designed. The construction cost is proportional to the volume, $50\ \text{€/m}^3$, while the revenue is proportional to the surface area, $200\ \text{€/m}^2$. (a) Write the net profit $G(r)$ as a function of the radius. (b) Find the radius that maximizes the profit.

Solution

(a) Sphere surface $S=4\pi r^2$, volume $V=\tfrac43\pi r^3$. The profit is revenue minus cost: $G(r)=200\cdot 4\pi r^2-50\cdot\tfrac43\pi r^3=800\pi r^2-\frac{200}{3}\pi r^3.$

(b) Differentiating and setting to zero: $G'(r)=1600\pi r-200\pi r^2=200\pi r(8-r)=0.$ The solutions are $r=0$ (trivial) and $r=8$. Since $G''(r)=1600\pi-400\pi r,\qquad G''(8)=1600\pi-3200\pi=-1600\pi<0,$ $r=8$ gives a maximum. $\boxed{\;r=8\ \text{m}\;}$