Question

The volume of a box(V) varies directly with its length(l). If a box in the group has a length of 30 inches, and the girth of 20 inches (perimeter of the side formed by the width and height), what is its height? Use k = 24. (Hint: Volume = length • width • height. Solve for length, and substitute into the equation for constant of proportionality.)?

Answer 1

$\bf \textit{V varies directly with l}\implies V=kl\qquad k=24\implies V=24l \\\\\\ \textit{volume of the box, or a rectangular prism}\\\\ V=lwh\qquad \begin{cases} l=length\\ w=width\\ h=height\\ ----------\\ k=24 \end{cases}\implies \boxed{lwh=24l} \\\\\\ \textit{girth of the box }\\\\ w+w+h+h=20\implies 2w+2h=20\implies 2(w+h)=20 \\\\\\ thus\qquad w+h=\cfrac{20}{2}\implies \boxed{w=10-h}$

$\bf \\\\ -------------------------------\\\\ lwh=24l\implies wh=24\implies (10-h)h=24\implies 10h-h^2=24 \\\\\\ 0=h^2-10+24\implies 0=(h-6)(h-4)\implies h= \begin{cases} 6\\ 4 \end{cases}$

now, notice, we didn't use the length of 30inches.... since the "l"'s cancel each other anyway, so it doesn't weight much on what the value for "h" is, by simply doing the substution of "w" from the Girth.