a grain silo consists of a cylindrical concrete tower surmounted by a metal hemispherical dome. the metal in…

Explanation:

Step1: Write volume formula

The volume of the silo $V = \pi r^{2}h+\frac{2}{3}\pi r^{3}$, and $V = 600$, so $\pi r^{2}h+\frac{2}{3}\pi r^{3}=600$, and we can express $h=\frac{600 - \frac{2}{3}\pi r^{3}}{\pi r^{2}}=\frac{600}{\pi r^{2}}-\frac{2}{3}r$.

Step2: Write cost - function

Let the cost per unit area of concrete be $1$, then the cost per unit area of metal is $2.5$. The surface - area of the cylinder is $A_{c}=2\pi rh$ and the surface - area of the hemisphere is $A_{h}=2\pi r^{2}$. The cost function $C = 2\pi rh+2.5\times2\pi r^{2}=2\pi r\left(\frac{600}{\pi r^{2}}-\frac{2}{3}r\right)+5\pi r^{2}$.

Step3: Simplify cost - function

$C=\frac{1200}{r}-\frac{4}{3}\pi r^{2}+5\pi r^{2}=\frac{1200}{r}+\left(5\pi-\frac{4}{3}\pi\right)r^{2}=\frac{1200}{r}+\frac{11}{3}\pi r^{2}$.

Step4: Differentiate cost - function

Differentiate $C(r)$ with respect to $r$: $C^\prime(r)=-\frac{1200}{r^{2}}+\frac{22}{3}\pi r$.

Step5: Set derivative equal to zero

Set $C^\prime(r) = 0$, then $-\frac{1200}{r^{2}}+\frac{22}{3}\pi r = 0$. Multiply through by $r^{2}$ to get $- 1200+\frac{22}{3}\pi r^{3}=0$. Then $\frac{22}{3}\pi r^{3}=1200$, and $r^{3}=\frac{3600}{22\pi}=\frac{1800}{11\pi}$.

Step6: Solve for $r$

$r=\sqrt[3]{\frac{1800}{11\pi}}\approx3.7$.

Answer:

$3.7$