Answer:
Explanation:
Given
volume [tex]V=16 ft^3[/tex]
Suppose base is square with side L
height of crate is h
Volume [tex]V=L^2\times h[/tex]
[tex]16=L^2\times h[/tex]
Cost of top and bottom area [tex]c_1=3L^2[/tex]
Cost of Side area [tex]c_2=4Lh\times 2=8Lh=8L\times \frac{16}{L^2}=\frac{128}{L}[/tex]
Total Cost [tex]C=c_1+c_2[/tex]
Total Cost [tex]C=3L^2+\frac{128}{L}[/tex]
Differentiate C w.r.t Length
[tex]\frac{dC}{dL}=6L-\frac{128}{L^2}[/tex]
[tex]L^3=\frac{128}{6}[/tex]
[tex]L=2.75 ft[/tex]
[tex]h=\frac{16}{2.75^2}=11.46 ft[/tex]
Dimensions are [tex]L\times L\times h=2.75\times 2.75\times 11.46[/tex]
