7.4k views
5 votes
Please show full solutions!!

Please show full solutions!!-example-1

1 Answer

3 votes

Answer:

diameter = height ≈ 6.3 m

Explanation:

You want the dimensions of the cylinder with minimum area that has a volume of 200 m³.

Minimum

The short answer is that a cylinder with minimum surface area has a height equal to its diameter. The volume is ...

V = (π/4)d²h = (π/4)d³

Then the diameter is ...

d = ∛(4V/π) = ∛(800/π) ≈ 6.3 . . . . meters

The height and diameter of the cylinder are about 6.3 meters.

Long answer

The volume relates the height to the diameter:

V = (π/4)d²h

h = (4V)/(πd²)

The surface area equation is ...

SA = (π/2)d² +πdh

SA = (π/2)d² +4V/d . . . . . surface area as a function of diameter

Setting the derivative to zero, we have ...

SA' = 0 = πd -4V/d² . . . . . . . derivative with respect to d

4V = πd³ . . . . . . . . . . . . . . . . multiply by d², add 4V

d = ∛(4V/π) = ∛(4·200/π) ≈ 6.3 . . . . as above

h = 4V/(πd²) = d³/d² = d . . . . . generic solution for minimum area

The height and diameter are about 6.3 m.

__

Additional comment

Rounded to the nearest tenth, the radius is 3.2 m.

The usual formulas for volume and area make use of the radius, r = d/2. Substituting d/2 for r in those formulas, we get formulas in terms of the diameter. Here, that is convenient because we know the diameter and height will have the same value.

<95141404393>

User Robusto
by
7.7k points