Chapter A.5, Problem 100E

(a)

To determine

Find what the given expression represents

Answer to Problem 100E

The expression $x^2(x-3)$ represents the volume of water.

Explanation of Solution

Given:

The figure shows a glass cube partially filled with water.

  

The given expression is $x^2\left(x-3\right)$

Consider the following expression:

  $x^2(x-3)$

A glass cube of side $x$ feet, height of a glass cube filled with water is $(x-3),$ and area of a base of a glass cube is $x^2$ .

Therefore, above expression represents a product of base area $x^2$ and height $(x-3)$

Because volume of a cube is,

length $\times$ breadth $\times$ height

Here length of a cube is $x$ feet, breadth of a cube is $x$ feet and height of a glass cube filled with water is $(x-3)$ feet

Above can be written as follows:

  $x^2(x-3)=x\times x\times (x-3)$

Hence, expression $x^2(x-3)$ represents the volume of water.

(b)

To determine

Find the capacity of the cube.

Answer to Problem 100E

The capacity of cube is $512{\text{ft}}^3$

Explanation of Solution

Given:

The figure shows a glass cube partially filled with water.

  

The given expression is $x^2\left(x-3\right)=320$

Consider the following equation:

  $x^2(x-3)=320$

To find capacity of the cube, factorize the above expression for the value of $x$

Above equation is factorable, factorize the equation by appropriate method.

Distribute $x^2$ over the equation $x^2(x-3)=320$

  $x^3-3x^2=320$

Subtract 320 from both sides:

  $x^3-3x^2-320=320-320$

Combine like terms to simplify the expression:

  $x^3-3x^2-320=\mathrm{0.......}(1)$

Put $x=8$ in the above expression.

  $\begin{array}{cc}{x}^3-3x^2-320& =0\\ 8^3-3\times 8^2-320& =0\\ 512-192-320& =0\\ 0& =0\end{array}$

Therefore $x=8$ is a zero of equation (1)

Hence above equation (1) can be written as follows:

  $(x-8)\left(x^2+5x+40\right)=\mathrm{0........}(2)$

Further solve the equation (2) $(x-8)\left(x^2+5x+40\right)=0$ by zero product rules

Set factor $(x-8)$ equal to zero:

  $x-8=0$

Simplify this expression for the value of $x$

  $x=8$

Further put another factor $x^2+5x+40=0$

Use quadratic formula to solve the above quadratic expression

  $\begin{array}{cc}x& =\frac{-5\pm \sqrt{5^2-160}}{2}\\ & =\frac{-5\pm \sqrt{25-160}}{2}\\ & =\frac{-5\pm \sqrt{-135}}{2}\end{array}$

This gives no real roots.

Hence the side length of cube is $x=8\text{ft}$

The volume of cube is product of base area and its height.

Since the base is square,

Area of base is $x^2$ and height is $x$

Therefore, the volume is $x^2\cdot x=x^3$

For $x=8$ the volume is given by

  ${(8)}^3=512$

Hence, the capacity of cube is $512{\text{ft}}^3$