Question

1.A box without a top is to be made from a rectangular piece of cardboard, with dimensions 8 in. by 10 in., by cutting out square corners with side length x and folding up the sides.(a)Write an equation for the volume V of the box in terms of x.(b)Use technology to estimate the value of x, to the nearest tenth, that gives the greatest volume. Explain your process.

Answer 1

Answer: The volume of the box is given by the formula:

(a) Equation of the box volume:

`\begin{gathered} V=l* w* h\rightarrow(1) \\ \\ l=\text{ Length} \\ \\ w=\text{ Width} \\ \\ h=\text{ Height} \end{gathered}`

According to the figure, we can rewrite the formula (1) as follows:

`\begin{gathered} l=(10-2x) \\ \\ w=(8-2x) \\ \\ h=x \\ \\ V(x)=(10-2x)(8-2x)x\rightarrow(2) \end{gathered}`

(b) The value of x for which the volume is the greatest is:

`\begin{gathered} \begin{equation*} V(x)=(10-2x)(8-2x)x \end{equation*} \\ \\ (dV(x))/(dx)=0 \\ \\ \\ (d)/(dx)[(10-2x)(8-2x)x]=0\rightarrow(3) \end{gathered}`

The solution to the equation (3) is as follows:

`\begin{gathered} \begin{equation*} (d)/(dx)[(10-2x)(8-2x)x] \end{equation*} \\ \\ \\ (d)/(dx)[4x^3-36x^2+80x]=0 \\ \\ \\ 12x^2-72x+80=0 \\ \\ \therefore\rightarrow \\ \\ \\ x=1.472in \\ \\ \\ \\ x=1.5in \end{gathered}`