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