Question

A silo is to be built by attaching a hemisphere roof onto a circular cylinder. The hemisphere part of the silo is constructed by a material costing $100 per square foot, and the sides are constructed with a different material costing $75 per square foot. The volume of the silo must be held to 5000 cubic feet. What radius and height, to two decimal places, will minimize the cost of the construction

Answer 1

The radius and height, in two decimal places are mathematically given as

r=10.60

What is the radius and height, in two decimal places?

Generally, the equation for the volume is mathematically given as

$V=\pi r^{2} h+\frac{2}{3} \pi r^{3}$

Therefore

$5000 &=\pi 0^{2} h+\frac{2}{3} \pi r^{3}$

$V(r, h) &=\pi r^{2} h+\frac{2}{3} \pi r^{3}-5000 \\\nabla V &=\left\langle 2 \pi r h+2 \pi r^{2}, \pi r^{2}\right\rangle$

$\cos t &=100\left(2 \pi r^{2}\right)+75(2 \pi r h) \\\\\cos t &=200 \pi r^{2}+150 \pi r h \\\\\nabla C(r, h) &=\langle 400 \pi r+150 \pi h, 150 \pi r\rangle$

Using Lagrange method

$\langle 400 \pi \gamma+150 \pi \gamma h, 150 \pi \gamma\rangle=\lambda\left\langle 2 \pi \gamma h+2 \pi \gamma^{2}, \pi r^{2}\right\rangle$

$400 \pi r+150 \pi r=\left(2 \pi r h+2 \pi r^{2}\right) \lambda$

$400 \pi \gamma+150 \pi h=300 \pi h+300 \pi r$

$400 \pi r-300 \pi r=150 \pi h$

$160 \pi r=150 \pi h \quad$

with

$h=\frac{2}{3} r$

$\pi r^{2} h+\frac{2}{3} \pi r^{3}=5000$

$\pi r}\left(\frac{2}{3} r \right)+\frac{2}{3} \pi^{3}=5000$

$\frac{4 \pi \gamma^{3}}{3}=5000 \quad$

$r^{3}=\left(\frac{5000 \times 3}{4 \pi}\right)$

r=10.60

In conclusion, the radius and height, in two decimal places is

r=10.60

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