a box is open and its length is twice its width. If 15000 cm of steel is used in making it, what is the maximum volume and what is its height?
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Standard equation for volume of the box is V = L*W*H
W = width of the box
L = length of the box = 2*W
H = height of the box
V = (2*W)*W*H = 2*W^2*H
Determine the relation for the surface area of the box.
The box is open so there is no top.
A1 = area of bottom = L*W = 2*W*W = 2*W^2
A2 = each end = W*H
A3 = each side = L*H = 2*W*H
Area = A1 + 2*A2 + 2*A3 = 2*W^2 + 2*W*H + 4*W*H = 2*W^2 + 6*W*H = 15000
Solve this for H in terms of W:
6*W*H = 15000 - 2*W^2
H = (15000 - 2*W^2)/(6*W) = (7500 - W^2)/(3*W)
Use this in the equation for the volume.
V = 2*W^2*H = 2*W^2*[(7500 - W^2)/(3*W)]
V = (2/3)*W*[7500 - W^2] = (2/3)*[7500*W - W^3]
Find dV/dW and set it equal to 0 to get the maximum.
dV/dW = (2/3)*[7500 - 3*W^2] = 0
7500 - 3*W^2 = 0
2500 - W^2 = 0
W = sqrt(2500) = 50
So the width for maximum volume is 50
Use this in the equation for volume to get the maximum volume.
Maximum volume = (2/3)*[7500*W - W^3] = (2*W/3)*[7500 - W^2]
V = (2*50/3)*5000 = 500000/3 = 166666.6667 cm^3
Height = (7500 - W^2)/(3*W) = 33.3333 cm
W = width of the box
L = length of the box = 2*W
H = height of the box
V = (2*W)*W*H = 2*W^2*H
Determine the relation for the surface area of the box.
The box is open so there is no top.
A1 = area of bottom = L*W = 2*W*W = 2*W^2
A2 = each end = W*H
A3 = each side = L*H = 2*W*H
Area = A1 + 2*A2 + 2*A3 = 2*W^2 + 2*W*H + 4*W*H = 2*W^2 + 6*W*H = 15000
Solve this for H in terms of W:
6*W*H = 15000 - 2*W^2
H = (15000 - 2*W^2)/(6*W) = (7500 - W^2)/(3*W)
Use this in the equation for the volume.
V = 2*W^2*H = 2*W^2*[(7500 - W^2)/(3*W)]
V = (2/3)*W*[7500 - W^2] = (2/3)*[7500*W - W^3]
Find dV/dW and set it equal to 0 to get the maximum.
dV/dW = (2/3)*[7500 - 3*W^2] = 0
7500 - 3*W^2 = 0
2500 - W^2 = 0
W = sqrt(2500) = 50
So the width for maximum volume is 50
Use this in the equation for volume to get the maximum volume.
Maximum volume = (2/3)*[7500*W - W^3] = (2*W/3)*[7500 - W^2]
V = (2*50/3)*5000 = 500000/3 = 166666.6667 cm^3
Height = (7500 - W^2)/(3*W) = 33.3333 cm