Optimization:You and your classmates have decided to enter an ice sculpting contest. Before you begin, you must create a block of ice with the following strict requirements:1. The volume must be 18 ft3.2. The length of the base must be three times the size of the width of the base.3. The long side of the base cannot be more than 12 ft long.4. The short side of the base must be at least 1 foot long. (There are no constraints of the height of the block.)Find the dimensions of the ice block with the largest possible surface are. Note that your team captain will require proof that you have, indeed, provided a block with the maximum surface area.

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Asked Oct 30, 2019
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Optimization:

You and your classmates have decided to enter an ice sculpting contest. Before you begin, you must create a block of ice with the following strict requirements:

1. The volume must be 18 ft3.

2. The length of the base must be three times the size of the width of the base.

3. The long side of the base cannot be more than 12 ft long.

4. The short side of the base must be at least 1 foot long. (There are no constraints of the height of the block.)

Find the dimensions of the ice block with the largest possible surface are. Note that your team captain will require proof that you have, indeed, provided a block with the maximum surface area. 

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Expert Answer

Step 1

Given an ice block with the following conditions,

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Volume 18 ft3 The length of the base must be three times the size of the width of the base The long side of the base cannot be more than 12 ft long The short side of the base must be at least 1 foot long. To Find: the dimensions of the ice block with the largest possible surface area

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Step 2

Now, let l be the length, w be the width and h be the height of ice block.

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Volume 18 l w h 18 The length of the base must be three times the size of the width of the base 1 3w W = 3 Substitute in the formula for volume: *h = 18 12h 18 3 12h 54 = 54 12

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Step 3

We calculate the su...

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S 2(lwlh hw) = 2 1*- lhh 3 4lh 12 = 2 3 3 (12 4lh) 3 54 2 1241 3 12 2 4. 55 3 216 2 12 2 /13216 3

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