Answer to Question #213536 in Civil and Environmental Engineering for Toylart

Question #213536

Use langrange multiplier to determine the dimension of rectangular box open at the top, having a volume of 32ft^3 and requiring the least amount of material for it's construction

Expert's answer

Let dimensions of the box be a, b and c feet respectively.

Rectangular box, open at the op

Given : "Volume = abc=32ft^3"

To find : least amount of material for its construction, which means minimum surface area.

Solution : "S=ab+2bc+2ca" is the surface area of the required box.

Thus, "V(a,b,c)=abc-32=0"

And "S(a,b,c)=ab+2bc+2ca" = minimum.

Using the Method of Lagrange's Multiplier, we get;

"\\triangledown S=\\lambda \\triangledown V"

"\\triangledown S=\\begin{bmatrix}\n S_a\\\\ S_b \\\\S_c\n\\end{bmatrix}=\\begin{bmatrix}\n b+2c\\\\ a+2c \\\\2a+2b\n\\end{bmatrix};" "\\triangledown V=\\begin{bmatrix}\n V_a\\\\ V_b \\\\V_c\n\\end{bmatrix}=\\begin{bmatrix}\n bc\\\\ ac \\\\ab\n\\end{bmatrix}"

"\\implies \\begin{bmatrix}\n b+2c\\\\ a+2c \\\\2a+2b\n\\end{bmatrix}=\\lambda \\begin{bmatrix}\n bc\\\\ ac \\\\ab\n\\end{bmatrix}"

"\\implies \\begin{bmatrix}\n b+2c\\\\ a+2c \\\\2a+2b\n\\end{bmatrix}= \\begin{bmatrix}\n \\lambda bc\\\\ \\lambda ac \\\\\\lambda ab\n\\end{bmatrix}"

"\\implies \\lambda = (b+2c)\/bc=(a+2c)\/ac=(2a+2b)\/ab"

"(b+2c)\/bc=(a+2c)\/ac \\implies 1+2c\/b=1+2c\/a"

"\\implies b=a"

"(a+2c)\/ac=(2a+2b)\/ab \\implies (a+2c)\/ac=""4a\/a^2=4\/a \\implies a\/c+2=4"

"\\implies c=a\/2"

"V=abc-32=a(a)(a\/2)-32=0 \\implies a^3-64 =0"

"\\implies a=4; b=4; c=2"

"\\implies S= ab+2bc+2ca=16+16+16=48 ft^2"

Thus,48 square feet is the least amount of material for its construction.

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