Chapter 12.1, Problem 3E

b.

To determine

Find the function V that represent the volume of the box.

Answer to Problem 3E

The equation of volume is $4(x^3+144x-24x^2)$ .

Explanation of Solution

Given:

It is given in the question that the it has to be create an open box from a square piece of material $24$ centimeters on a side. If it is cut into equal squares with sides of length x from the corners and turn up the sides.

Concept Used:

In this, use the concept of understanding the question and make a linear equation and also use the formula of volume i,e, $V=l\cdot b\cdot h$ .

Calculation:

Since, the size of the square piece = $24\times 24c{m}^2$ .

Let say size of square cut from corner = $x\times xc{m}^2$

The sides of the Open box would be $24-2x,24-2x\&x$ .

Volume of the Open box = $(24-2x)(24-2x)(x)$

  $\begin{array}{l}$ =x{(24-2x)}^2$$ =x\cdot 2^2{(12-x)}^2$=4x(x^2+144-24x)\\ =4(x^3+144x-24x^2)\end{array}$

Conclusion:

The equation of volume is $4(x^3+144x-24x^2)$ .

c.

To determine

Calculate the volume by using the different values of the number in the table.

Answer to Problem 3E

The values for the different x are $948,1011.48,1023.48,1024,1023.52,1012.48,980$ .

Explanation of Solution

Given:

It is given in the question that the box has a maximum volume when $x=4$ and the the given values are $3,3.5,3.9,4,4.1,4.5,5$ .

Concept Used:

In this, use the concept of putting the given values in the above get solution and get the answer.

Calculation: Here, the equation is $4(x^3+144x-24x^2)$ .

  $V(x)=4(x^3+144x-24x^2)$

Put $x=3$ ,

  $V(3)=4(3^3+144\times 3-24\times 3^2)=4(27+504-294)=948$

Put $x=3.5$ ,

  $V(3.5)=4({3.5}^3+144\times 3.5-24\times {3.5}^2)=4(42.87+504-294)=1011.48$

Put $x=3.9$ ,

  $V(3.9)=4({3.9}^3+144\times 3.9-24\times {3.9}^2)=4(59.31+561.6-365.04)=1023.48$

Put $x=4$ ,

  $V(4)=4(4^3+144\times 4-24\times 4^2)=4(64+576-384)=1024$

Put $x=4.1$ ,

  $V(4.1)=4({4.1}^3+144\times 4.1-24\times {4.1}^2)=4(68.92+590.4-403.44)=1023.52$

Put $x=4.5$ ,

  $V(4.5)=4({4.5}^3+144\times 4.5-24\times {4.5}^2)=4(91.12+648-486)=1012.48$

Put $x=5$ ,

  $V(5)=4(5^3+144\times 5-24\times 5^2)=4(125+720-600)=980$

Conclusion:

The values are $948,1011.48,1023.48,1024,1023.52,1012.48,980$ .

d.

To determine

Draw the graph of the function.

Explanation of Solution

Given:

It is given in the question that the expression for the volume is $V=4x(x^2+144-24x)$ .

Graph:

  

Interpretation:

Here, the function is $V=4x(x^2+144-24x)$ , put that function in the graphing calculator, it gives the right graph in proper way.

In the graph, it has to be seen that the maximum volume is at $x=4$ .