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3 years ago

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A box with a square base and open top must have a volume of 13,500 cm3. Find the dimensions of the box that minimize the amount

of material used

1 answer:

Alex17521 [72]3 years ago

5 0

Answer:

Step-by-step explanation:

Let the side of the square base be x

h be the height of the box

Volume V = x²h

13500 = x²h

h = 13500/x² ..... 1

Surface area = x² + 2xh + 2xh

Surface area S = x² + 4xh ...... 2

Substitute 1 into 2;

From 2; S = x² + 4xh

S = x² + 4x(13500/x²)

S = x² + 54000/x

To minimize the amount of material used; dS/dx = 0

dS/dx = 2x - 54000/x²

0 = 2x - 54000/x²

0 = 2x³ - 54000

2x³ = 54000

x³ = 27000

x = ∛27000

x = 30cm

Since V = x²h

13500 = 30²h

h = 13500/900

h = 15cm

Hence the dimensions of the box that minimize the amount of material used is 30cm by 30cm by 15cm

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If two angles are complementary, then the value of tangent of one angle is equal to the cotangent of the other angle

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