Answer:
The dimensions of the box are:
W=2.97 ft
L = 8.91ft
H=2.08ft
Step-by-step explanation:
Volume of the box= 55ft³
LWH=55
Length of the base, L = 3W
3W*WH=55
3W²H=55
[TeX]H= \frac{55}{3W^2}[/TeX]
Metal costs =$7/ft²
Wood costs =$5/ft².
Area of the Sides= 2(LH+WH)
=[TeX]\frac{2*3W*55}{3W^2}+\frac{2*W*55}{3W^2}\\=\frac{110}{W}+\frac{110}{3W}[/TeX]
Cost of the sides therefore is:
[TeX]5(\frac{110}{W}+\frac{110}{3W})\\=\frac{550}{W}+\frac{550}{3W}[/TeX]
Area of the top and bottom=2LW=2*3W*W=6W²
Cost=7*6W²=42W²
Total Cost, C(W) of the box:
[TeX]=\frac{550}{W}+\frac{550}{3W}+42W^2\\=\frac{1650+550+126W^3}{3W}\\=\frac{2200+126W^3}{3W}[/TeX]
To find the minimum cost, we set the derivative of C(W) to be equal to zero.
[TeX]C^{1}W=\frac{-2200+84W^3}{3W^2}[/TeX]
[TeX]\frac{-2200+84W^3}{3W^2}=0[/TeX]
84W³=2200
W³=26.19
W=2.97 ft
L = 3W=3*2.97=8.91ft
[TeX]H= \frac{55}{3X2.97^2}=2.08ft[/TeX]
The dimensions of the box are:
W=2.97 ft
L = 8.91ft
H=2.08ft
