Mathematics: A box with a square base and open top must have a volume of 256000 c m 3 . We wish to find the dimensions of the box that minimize the amount of material used. First, find a formula for the surface area of the box in terms of only x , the length of one side of the square base.

A box with a square base and open top must have

Ответ:

jordan495413

16.07.2021

Mathematics

Follows are the response to the given question:

Step-by-step explanation:

The volume of the box:

$V = x\times x \times h = 256000 \ cm^3\\\\\to x^2 \times h = 256000\\\\\to h = \frac{256000}{x^2}$

The surface area of the open box is:

$A(x) = x \times x + 2 \times (x \times h +x \times h)\\\\A(x) = x^2 + 4 \times x \times h\\\\A(x) = x^2 + \frac{1024000}{x}\\\\\frac{d(x^n)}{dx} = n \times x^{(n - 1)}\\\\$

Use above formula

$A'(x) = 2 \times x - \frac{1024000}{x^2}\\\\$

$A'(x) = 0\\\\2\times x - \frac{1024000}{x^2} = 0\\\\2x = \frac{1024000}{x^2}\\\\x^3 = 512000\\\\x = (512000)^{(\frac{1}{3})} = 80\ cm\\\\$

Now

$A''(x) = 2\times 1 + 2\times \frac{1024000}{x^3}\\\\A''(x) = 2 + \frac{2048000}{x^3}\\\\x = 80 \ cm\\\\A''(80) = 2 + \frac{2048000}{80^3} = 6\\\\$

therefore A"(x) 0, x amount of material used in minimum.

$h = \frac{256000}{80^2} = 40\ cm$

A box with a square base and open top must have a volume of 256000 c m 3 . We wish to find the dimen

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Ответ:

ash9951

26.07.2020

Mathematics

just copy the question and paste it on the internet.

Step-by-step explanation:

to copy hold control which is the the button that says ctrl and press,you now copied the question.

i forgot to say highlight the question,you should know how to do that.

to paste go on google or whatever you use,and hold control button and press v, and you now pasted the question on the internet.

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A box with a square base and open top must have a volume of 256000 c m 3 . We wish to find the dimensions of the box that minimize the amount of material used. First, find a formula for the surface area of the box in terms of only x , the length of one side of the square base.

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