Answer:
Length = 47 in
Radius = 47/π in
Step-by-step explanation:
Let 'h' be the length of the package, and 'r' be the radius of the cross section.
The length and girth combined are:
[tex]L+G=141=h+2\pi r\\h=141-2\pi r[/tex]
The volume of the cylindrical package is:
[tex]V=A_b*h\\V=\pi r^2*h[/tex]
Rewriting the volume as a function of 'r':
[tex]V=\pi r^2*h\\V=\pi r^2*(141-2\pi r)\\V=141\pi r^2-2\pi^2 r^3[/tex]
The value of 'r' for which the derivate of the volume function is zero yields the maximum volume:
[tex]V=141\pi r^2-2\pi^2 r^3\\\frac{dV}{dr}=282\pi r-6\pi^2r^2=0\\ 6\pi r=282\\r=\frac{47}{\pi} \ in[/tex]
The length is:
[tex]h=141-2\pi r=141-2\pi*\frac{47}{\pi}\\h=47\ in[/tex]
The dimensions that yield the maximum volume are:
Length = 47 in
Radius = 47/π in
