Question

2. A more efficient packing of the discs is obtained by dividing the metal sheet into hexagons and cutting the circular lids and bases from the hexagons (see the last figure). Show that if this strategy is adopted, then
$\frac{h}{r}=\frac{4 \sqrt{3}}{\pi} \approx 2.21$

Answer 1

This exercise is about optimization and seeks to prove that if the new strategy of packing the discs is adopted, then h/r = $\sqrt[4]{3}$/n ≈2.21.

What is the proof for the above strategy?

We must determine the amount of metal consumed by each end, or the area of each hexagon.

The hexagon is divided into six congruent triangles, each of which has one side (s in the diagram) in common with the hexagon.

Step I

Next, let's derive the length of s = 2r tan π/6 = (2/($\sqrt{3}$)r². From this we can state that the area of each of the triangles are 1/2(sr) = (1/$\sqrt{3}$)r²

while the total area of the hexagon is 6 * (1/$\sqrt{3}$)r² = (2/$\sqrt{3}$)r².

From the above, we can state that the quantity we want to minimize is given as:

A =  2πrh + 2* (2/$\sqrt{3}$)r²

Step 2

Next, we substitute for h and differentiate. This gives us:

da/dr = - (2V/r²) + $\sqrt[8]{3r}$.

Let us equate the above to zero.

$\sqrt[8]{3r} ^{3}$ = 2V = 2πr²h ⇒  h/r =$\sqrt[4]{3}$/n  

The above is approximately 2.21

Because d²A/dr²=$\sqrt[8]{3}$ + 4V/r$^{3}$ > 0 the above minimizes A.

Learn more about Optimization at:

brainly.com/question/17083176

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