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timama [110]
4 years ago
8

A box with a square base and open top must have a volume of 4,000 cm3. Find the dimensions of the box that minimize the amount o

f material used.
Mathematics
1 answer:
Marysya12 [62]4 years ago
5 0
Let the side of the square base be x, and the height of the box be h. 

<span>The material of the base is x^2, and the material of the four sides is 4xh. </span>

<span>4000 = hx^2 </span>
<span>h = 4000/x^2 </span>

<span>The total material is </span>

<span>M = x^2 + 4x(4000/x^2) = x^2 + 16000/x </span>

<span>Take the first derivative of M and set equal to 0. </span>

<span>M' = 2x - 16000/x^2 = 0 </span>

<span>Multiply by x^2: </span>

<span>2x^3 = 16000 </span>
<span>x^3 = 8000 </span>
<span>x = 20; h = 10; M = 1200</span>
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