
Calc Word Problems2
Some airlines have restrictions on the size of items of luggage that passengers are allowed to take with them. Suppose that one has a rule that the sum of the length, width and height of any piece of luggage must be less than or equal to 150 cm. A passenger wants to take a box of the maximum allowable volume.
If the length and width are to be equal, what should the dimensions be? In this case, what is the volume?
If the length is be twice the width, what should the dimensions be?
In this case, what is the volume?

a)$\displaystyle l=w$
$\displaystyle 2l+h=150$
$\displaystyle h=1502l$
$\displaystyle h>0 \Rightarrow l\in(0,75)$
$\displaystyle v=lwh \Rightarrow v(l)=l^2(1,52l)=2l^3+1,5l^2$
$\displaystyle v'(l)=6l^2+3l=3l(12l)=0 \Rightarrow l=0,5 m=50cm$
b)$\displaystyle l=2w$
$\displaystyle 2w+h=150$
$\displaystyle h=1503w$
$\displaystyle h>0 \Rightarrow w\in(0,50)$
$\displaystyle v=lwh \Rightarrow v(w)=2w^2(1,53w)=6w^3+3w^2$
$\displaystyle v'(w)=18w^2+6w=6w(13w)=0 \Rightarrow w=\frac{1}{3} m=33\frac{1}{3}cm$