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lions [1.4K]
2 years ago
9

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
Mathematics
1 answer:
valkas [14]2 years ago
4 0

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.

<h3>What is the proof for the above strategy?</h3>

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 <em>h</em><em> </em> 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

#SPJ1

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Missing data:

Viewer's Age Group  Excellent Good Average Poor Marginal Total

16–25                          52                 42       12       7           113

26–35                          33                50         5       9          97

36–45                          58                 12       28      34         132

46–55                          25                17       22       12            76

56 +                               12                 5          3         8             28

Marginal Total             180               126      70      70           446

A rating of good or excellent indicates the audience liked the movie, while a rating of poor indicates the audience disliked the movie.

<h3>How to determine the rating of the film from the 46–55 age group?</h3>

A movie producer accomplished a survey behind a preview screening of her latest movie to estimate how the film would be accepted by viewers from various age groups. The table displays the numbers of viewers in various age groups who ranked the film excellent, good, average, and poor.

25/446 = 0.05605

0.05605 * 100% = 5.605%

Out of the entire respondents, the percentage of respondents from the 46–55 age group who ranked the film excellent exists at 5.605%.

To learn more about data refer to:

brainly.com/question/4219149

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