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, $600=\pi r^{2}h+\frac{2}{3}\pi r^{3}$, and we can express $h$ in terms of $r$ as $h=\frac{600 - \frac{2}{3}\pi r^{3}}{\pi r^{2}}=\frac{600}{\pi r^{2}}-\frac{2}{3}r$.

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 cylindrical part is $A_{c}=2\pi rh$ and the surface - area of the hemispherical part is $A_{h}=2\pi r^{2}$. The cost function $C = 2\pi rh+2.5\times(2\pi r^{2})=2\pi r\left(\frac{600}{\pi r^{2}}-\frac{2}{3}r\right)+5\pi r^{2}$. Simplify the cost function: [ \begin{align*} 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} \end{align*} ]

Step3: Differentiate the cost function

Differentiate $C(r)$ with respect to $r$: $C^\prime(r)=-\frac{1200}{r^{2}}+\frac{22}{3}\pi r$. Set $C^\prime(r) = 0$ to find the critical points: [ \begin{align*} -\frac{1200}{r^{2}}+\frac{22}{3}\pi r&=0\ \frac{1200}{r^{2}}&=\frac{22}{3}\pi r\ 1200\times3&=22\pi r^{3}\ r^{3}&=\frac{3600}{22\pi}=\frac{1800}{11\pi}\ r&\approx3.7 \end{align*} ]

Step4: Find the height

Substitute $r = 3.7$ into the formula for $h$: [ \begin{align*} h&=\frac{600}{\pi\times(3.7)^{2}}-\frac{2}{3}\times3.7\ &=\frac{600}{\pi\times13.69}-\frac{7.4}{3}\ &\approx\frac{600}{43.08}-\frac{7.4}{3}\ &\approx13.9 - 2.5\ &\approx11.4 \end{align*} ]

Answer:

$11.4$