Chapter 4.3, Problem 43E

(a)

To determine

To Find: The polynomial function to model the situation.

Answer to Problem 43E

The required polynomial function is $V\left(x\right)=x^3+12x^2-47x+60$ .

Explanation of Solution

Given:

The block of ice is 4 feet by 5 feet by 5feet.

Calculation:

Consider the volume of the ice block is obtained as,

  $\begin{array}{c}V\left(x\right)=\left(3-x\right)\left(4-x\right)\left(5-x\right)\\ =\left(12-7x+x^2\right)\left(5-x\right)\\ =\left(60-12x-35x+7x^2+5x^2-x^3\right)\\ =x^3+12x^2-47x+60\end{array}$

(b)

To determine

To Find: Thegraph of the function.

Answer to Problem 43E

The graph of the function is shown in Figure 1

Explanation of Solution

The graph for the function is $V\left(x\right)=x^3+12x^2-47x+60$ .

The graph of the function is shown in Figure 1

  

Figure 1

(c)

To determine

To Find: The equation to model the situation such that the volume of the ice is reduced to $\frac{3}{5}$ of the original volume and then write the equation to model the situation.

Answer to Problem 43E

The required function is $-x^3+12x^2-47x+60=36$ .

Explanation of Solution

Consider the volume of the ice block is,

  $\begin{array}{c}V=\left(3\right)\left(4\right)\left(5\right)\\ =60{\text{ft}}^2\end{array}$

Since, the volume is reduced by $\frac{3}{5}$ . Then,

  $\begin{array}{l}-x^3+12x^2-47x+60=\frac{3}{5}\left(60\right)\\ -x^3+12x^2-47x+60=36\end{array}$

(d)

To determine

To Find: The amount that must be taken from each of the dimensions.

Answer to Problem 43E

The amount of ice that should be taken from each of the side is 0.6 feet.

Explanation of Solution

Consider the equation from part b is,

  $-x^3+12x^2-47x+60=36$

Then,

  $\begin{array}{l}-x^3+12x^2-47x+24=0\\ x^3-12x^2+47x-24=0\end{array}$

Then, the graph of the equation is shown in Figure 2

  

Figure 2

From the graph the amount of ice that should be taken from each of the side is 0.6 feet.