Chapter 4.5, Problem 49E

To determine

To find: the dimensions of the mold.

Answer to Problem 49E

The length of mold is 4 cm, width is 4 cm and height is $4+3=7$ cm.

Explanation of Solution

Given information:

Each mold is a rectangular prism with a height 3 centimeters greater than the length of each side of its square base. Each mold holds 112 cubic centimeters of glass.

Calculation:

Let the length of one the square base is $x$ . So, the height is $x+3$ .

The volume of the glass is,

  $\begin{array}{c}V=x\cdot x\cdot \left(x+3\right)\\ 112=x^2\left(x+3\right)\\ 112=x^3+3x^2\\ x^3+3x^2-112=0\end{array}$

Now solve the equation for $x$

List the possible rational solutions.

The leading coefficient 1 and the constant term is $-112$ . So, the possible rational solutions are,

  $\begin{array}{c}x=\frac{\pm 1,\pm 2,\pm 4,\pm 7,\pm 8,\pm 28,\pm 56,\pm 112}{\pm 1}\\ =\pm 1,\pm 2,\pm 4,\pm 7,\pm 8,\pm 28,\pm 56,\pm 112\end{array}$

Test possible solutions using synthetic division until s solution is found.

For $x=4$ ,

  $\begin{array}{ccccc}4& 1& 3& 0& -112\\ & & 4& 28& 112\\ & 1& 7& 28& 0\end{array}$

So, one factor is $x-4$ .

Factor completely using the result of synthetic division.

Writing as a product of factors,

  $\begin{array}{c}{x}^3+3x^2-112=0\\ \left(x-4\right)\left(x^2+7x+28\right)=0\\ x-4=0or{x}^2+7x+28=0\\ x=4or{x}^2+7x+28=0\end{array}$

Because the discriminant of $x^2+7x+28=0$ is negative, so this equation has no real solution.

Therefore, the length of mold is 4 cm, width is 4 cm and height is $4+3=7$ cm.