Chapter 4.6, Problem 47E

To determine

To find: the value of $x$ .

Answer to Problem 47E

The value of $x$ is 4.26ft.

Explanation of Solution

Given information:

A solid monument with the dimensions shown is to be built using 1000 cubic feet of marble.

  

Calculation:

The volume of the monument is

  $V=x{\left(2x+6\right)}^2+\frac{1}{3}x{\left(2x\right)}^2$

Because the volume is 1000 cubic feet,

  $\begin{array}{c}x{\left(2x+6\right)}^2+\frac{1}{3}x{\left(2x\right)}^2=1000\\ 12x^3+72x^2+108x+4x^3=3000\\ 16x^3+72x^2+108x-3000=0\end{array}$

Because negative real and imaginary root doesn’t make sense, so need to check for only positive real root.

By the Descartes’s rule, the equation has one positive real root.

Input function $Y_1=16X^3+72X^2+108X-3000$ using key

  $16X\wedge 3+72X{x}^2+108X-3000$

The display is shown below:

  

Press the GRAPH key

  

To find the zero, press 2^ndTRACE 2 ENTER.

  

From the above display, the function has two zeros at approximately $x=4.26$ .

Therefore the value of $x$ is 4.26ft.