Lee has a small amount of money to spend in metal sheet. He wants to make a open box to store 20 m^3 of water. The height of the box is 20 cm. What should the dimensions be to minimize the cost.

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Lee wants to a construct an open metal box to store 20 m^3 of water. The height of the box should be 20 cm or 0.2 m. Let the base of the box have dimensions x and y.

As the box has a capacity 20 m^3, 0.2*x*y = 20. The...

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Lee wants to a construct an open metal box to store 20 m^3 of water. The height of the box should be 20 cm or 0.2 m. Let the base of the box have dimensions x and y.

As the box has a capacity 20 m^3, 0.2*x*y = 20. The surface area of the box is x*y + 2*0.2*x + 2*0.2*y

0.2*x*y = 20

=> y = 100/x

Substitute this in the formula for surface area of the box,

SA = x*y + 2*0.2*x + 2*0.2*y = x*(100/x) + 2*0.2*x + 2*0.2*(100/x)

= 100 + 0.4*x +40/x

To reduce SA, solve `(dSA)/(dx) = 0`

=> `0.4 - 40/x^2 = 0`

=> `x^2 = 40/0.4 = 100`

=> x = 10

y = 10

The dimensions of the box to reduce the amount of metal used is 0.2 m x 10 m x 10 m

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