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Zolol [24]
3 years ago
12

A storage box with a square base must have a volume of 90 cubic centimeters. the top and bottom cost $0.60 per square centimeter

and the sides cost $0.30 per square centimeter. find the dimensions that will minimize cost. (let x represent the length of the sides of the square base and let y represent the height.
Mathematics
2 answers:
Assoli18 [71]3 years ago
8 0

Answer:

side length of 3.56 cm and height of 7.10 cm

Step-by-step explanation:

Paul [167]3 years ago
4 0

<u>Answer-</u>

<em>For </em><em>side length of 3.56 cm</em><em> and </em><em>height of 7.10 cm</em><em> the cost will be minimum.</em>

<u>Solution-</u>

Let us assume that,

x represents the length of the sides of the square base,

y represent the height.

Given the volume of the box is 90 cm³, so

\Rightarrow V=90\\\\\Rightarrow x^2\times y=90\\\\\Rightarrow y=\dfrac{90}{x^2}

As the top and bottom cost $0.60 per cm² and the sides cost $0.30 per cm². Total cost C will be,

C=\text{cost for top and bottom}+\text{cost for rest 4 sides}\\\\=(2x^2\times 0.6)+(4xy\times 0.3)\\\\=(2x^2\times 0.6)+(4x\times \frac{90}{x^2}\times 0.3)\\\\=1.2x^2+ \dfrac{108}{x}

Then,

C'=\dfrac{d}{dx}(1.2x^2+ \dfrac{108}{x})=2.4x-\dfrac{108}\\\\C''=\dfrac{d^2}{dx^2}(1.2x^2+ \dfrac{108}{x})=2.4+\dfrac{216}{x^3}

As C'' has all positive terms  so, for every positive value of x (as length can't be negative), C'' is positive.

So, for minima C' = 0

\Rightarrow 2.4x-\dfrac{108}{x^2}=0\\\\\Rightarrow 2.4x=\dfrac{108}{x^2}\\\\\Rightarrow x^3=\dfrac{108}{2.4}=45\\\\\Rightarrow x=3.56

Then,

y=\dfrac{90}{x^2}

y=\dfrac{90}{3.56^2}

y=7.10

Therefore, for side length of 3.56 cm and height of 7.10 cm the cost will be minimum.

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