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3 years ago

What is the answer for #7?

Mathematics

1 answer:

Karo-lina-s [1.5K]3 years ago

3 0

Answer:

radius: 1.84 in
height: 3.68 in

Step-by-step explanation:

After you've worked a couple of "optimum cylinder" problems, you find that the cylinder with the least surface area for a given volume has a height that is equal to its diameter. So, the volume equation becomes ...

V = πr²·h = 2πr³ = 39 in³

Then the radius is ...

r = ∛(39/(2π)) in ≈ 1.83779 in ≈ 1.84 in

h = 2r = 3.67557 in ≈ 3.68 in

_____

The total surface area of a cylinder is ...

S = 2πr² + 2πrh

For a given volume, V, this becomes ...

S = 2π(r² +r·(V/(πr²))) = 2πr² +2V/r

The derivative of this with respect to r is ...

S' = 4πr -2V/r²

Setting this to zero and multiplying by r²/2 gives ...

0 = 2πr³ -V

r = ∛(V/(2π)) . . . . . . . . looks a lot like the expression above for r

If we substitute the equation for V into the equation just above this last one, we have ...

0 = 2πr³ - πr²·h

Dividing by πr² gives ...

0 = 2r - h

h = 2r . . . . . generic solution for cylinder optimization problems

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Therefore, the probability is P=9/400.

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