A three-dimensional space's occupied volume is measured. It is frequently expressed as a numerical value using SI-derived units, other imperial units, or US customary units. Volume definition and length definition are connected.

The Volume of a box with a square base x by x cm and height h cm is

$V = x^2h$

Since the surface area directly affects the amount of material used, we can reduce the amount of material by reducing the surface area.

The surface area of the box described is $A = x^2 + 4xh$

We need A as a function of x alone, so we'll use the fact that

$V = x^2h= 70304 cm^3$

⇒ $h = \frac{70304}{x^2}$

So, the area becomes,

$A = x^2+4x(\frac{70304}{x^2})\\ A = x^2 + \frac{281216}{x}$

We want to minimize A, so

$A' = 2x - \frac{281216}{x^2} =0\\\frac{2x^3-281216}{x^2}=0\\ {2x^3-281216} = 0\\x^3 - 140608=0\\x^3 = 140608\\x = 52$

The second derivative test verifies that A has a minimum at this critical number:

$A'' = 2+\frac{562432}{x^3}$ which is positive at x = 52.

Now,

$h = \frac{70304}{x^2}= \frac{70304}{(52)^2}=26$

Hence, The box should have base 52 cm by 52 cm and height 26 cm.

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8 0

1 year ago

What is one astronomical unit in inches

Aneli [31]

Answer:

5.89E+12

Step-by-step explanation:

8 0

3 years ago