A shipping company must design a closed rectangular

Ответ:

iiMxlissaii

23.01.2022

At minimum cost, Length = 8 foot, Height = 18 foot

Step-by-step explanation:

Volume of the closed rectangular shipping crate = 1152ft^3

The crate has a square base from the given statement.

Therefore: Volume = Square Base Area X Height = l^2h

l^2h =1152

$h=\dfrac{1152}{l^2}$

Total Surface Area of a Cuboid =2(lb+lh+bh)

Since we have a square base

Total Surface Area =2(l^2+lh+lh)

The Total Surface Area of the Crate

= Area of Top + Area of Bottom + Area of Sides

=l^2+l^2+4lh

The material for the top and sides costs $2 per square foot and the material for the bottom costs $7 per square foot.

Therefore, Cost of the Material for the Crate

$C=7l^2+2(l^2+4lh)\\C=9l^2+8lh$

Recall earlier that we derived: $h=\dfrac{1152}{l^2}$

Substituting into the formula for the Total Cost

$C=9l^2+8l(\dfrac{1152}{l^2})\\C=9l^2+\dfrac{9216}{l}\\C=\dfrac{9l^3+9216}{l}$

The minimum costs for the material occurs at the point where the derivative equals zero.

$C^{'}=\dfrac{18l^3-9216}{l^2}=0\\18l^3-9216=0\\18l^3=9216\\l^3=512\\l=\sqrt[3]{512}=8$

Recall:

$h=\dfrac{1152}{l^2}=\dfrac{1152}{8^2}=18$

The dimensions that will minimize the cost of the box are: Length = 8 foot, Height = 18 foot

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