A packaging company creates different sized cardboard boxes. The volume of a box is given by $${\mathtt{v}} = {\mathtt{18}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{3}}}{\mathtt{\,-\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{45}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{\mathtt{5}}$$
a)Determine expressions for the possible dimensions of these boxes.
b)Determine the dimensions and volume if x-2cm
Thanks in advance
+118723
$$\\v=18x^3-2x+45x^2-5\\\\$$
$${factor}{\left({\mathtt{18}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{3}}}{\mathtt{\,-\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{45}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{\mathtt{5}}\right)} = \left({\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{5}}\right){\mathtt{\,\times\,}}\left({\mathtt{3}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,-\,}}{\mathtt{1}}\right){\mathtt{\,\times\,}}\left({\mathtt{3}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}\right)$$
The dimensions of this box are 2x+5, 3x-1 and 3x+1
b) do you mean x=2
$${\mathtt{18}}{\mathtt{\,\times\,}}{{\mathtt{2}}}^{{\mathtt{3}}}{\mathtt{\,-\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{2}}{\mathtt{\,\small\textbf+\,}}{\mathtt{45}}{\mathtt{\,\times\,}}{{\mathtt{2}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{\mathtt{5}} = {\mathtt{315}}$$
$${\mathtt{315}} = \left({\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{5}}\right){\mathtt{\,\times\,}}\left({\mathtt{3}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,-\,}}{\mathtt{1}}\right){\mathtt{\,\times\,}}\left({\mathtt{3}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}\right) \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{\left({\sqrt{{\mathtt{551}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,\small\textbf+\,}}{\mathtt{27}}\right)}{{\mathtt{12}}}}\\
{\mathtt{x}} = {\frac{\left({\sqrt{{\mathtt{551}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,-\,}}{\mathtt{27}}\right)}{{\mathtt{12}}}}\\
{\mathtt{x}} = {\mathtt{2}}\\
\end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}\left({\frac{{\mathtt{9}}}{{\mathtt{4}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1.956\: \!115\: \!765\: \!718\: \!968\: \!9}}{i}\right)\\
{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{{\mathtt{9}}}{{\mathtt{4}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1.956\: \!115\: \!765\: \!715\: \!987\: \!5}}{i}\\
{\mathtt{x}} = {\mathtt{2}}\\
\end{array} \right\}$$
The only real solution is x=2
The dimensions are 9, 5 and 7 cm
+118723
$$\\v=18x^3-2x+45x^2-5\\\\$$
$${factor}{\left({\mathtt{18}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{3}}}{\mathtt{\,-\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{45}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{\mathtt{5}}\right)} = \left({\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{5}}\right){\mathtt{\,\times\,}}\left({\mathtt{3}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,-\,}}{\mathtt{1}}\right){\mathtt{\,\times\,}}\left({\mathtt{3}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}\right)$$
The dimensions of this box are 2x+5, 3x-1 and 3x+1
b) do you mean x=2
$${\mathtt{18}}{\mathtt{\,\times\,}}{{\mathtt{2}}}^{{\mathtt{3}}}{\mathtt{\,-\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{2}}{\mathtt{\,\small\textbf+\,}}{\mathtt{45}}{\mathtt{\,\times\,}}{{\mathtt{2}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{\mathtt{5}} = {\mathtt{315}}$$
$${\mathtt{315}} = \left({\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{5}}\right){\mathtt{\,\times\,}}\left({\mathtt{3}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,-\,}}{\mathtt{1}}\right){\mathtt{\,\times\,}}\left({\mathtt{3}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}\right) \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{\left({\sqrt{{\mathtt{551}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,\small\textbf+\,}}{\mathtt{27}}\right)}{{\mathtt{12}}}}\\
{\mathtt{x}} = {\frac{\left({\sqrt{{\mathtt{551}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,-\,}}{\mathtt{27}}\right)}{{\mathtt{12}}}}\\
{\mathtt{x}} = {\mathtt{2}}\\
\end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}\left({\frac{{\mathtt{9}}}{{\mathtt{4}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1.956\: \!115\: \!765\: \!718\: \!968\: \!9}}{i}\right)\\
{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{{\mathtt{9}}}{{\mathtt{4}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1.956\: \!115\: \!765\: \!715\: \!987\: \!5}}{i}\\
{\mathtt{x}} = {\mathtt{2}}\\
\end{array} \right\}$$
The only real solution is x=2
The dimensions are 9, 5 and 7 cm
