Question

2. Volume 1 hox of maximum volume is to be made from a repor material 24 centimeters on a side onlig plans from the comers and turning up (a) Write the volume V as a function of the length of the comer squares. What is the domain of the function? A) Use a graphing utility to graph the volume function and approximate the dimensions of the box that yield a maximum volume.

Answer 1

$\begin{gathered} a)V(x)=x(24-2x)^2 \\ b)\text{ Dimension for max i}mum\text{ volume:} \\ (x,\text{ V(x)) = (4, 102}4) \end{gathered}$

Explanation:

$\begin{gathered} \text{The base of the box is a square:} \\ \text{Area of square = length}^2 \\ \text{ length = 24 - 2x} \\ \text{Area of square = }(24-2x)^2 \end{gathered}$$\begin{gathered} \text{Volume of the box = Area of square }\times\text{ height = l}\times w\times h \\ h\text{ }=x \\ \text{Volume of the box = }(24-2x)^2(x) \\ \text{V}(x)\text{= }x(24-2x)^2 \end{gathered}$$\begin{gathered} \text{Domain: are the x values of the function} \\ the\text{ dimensions of the box cannot be negative},\text{ x will be greater than 0} \\ x\text{ > 0} \\ 24\text{ - 2x > 0} \\ 24\text{ > 2x} \\ 12\text{ > x} \\ x\text{ < 12} \\ \text{Domain: 0 < x < 12} \end{gathered}$$\begin{gathered} b)\text{ using graphi}ng\text{ calculator:} \\ x\text{ = length, y = volume } \\ \text{The max i}mum\text{ volume is 1024 at x = 4} \\ (x,\text{ V(x)) = (4, 102}4) \end{gathered}$