Chapter 11.1, Problem 5E

a.

To determine

To check the volume V of the box is $4x{(12-x)}^2$ .

Explanation of Solution

Given: Length of the open square box = 24 cm

Length of the open square box = 24cm

It cut into equal squares so,

Let each side of the square cut off from each corner be  x  cm

Then the base of the box will be of side 18- 2x  cm and the height of the box will be  x  cm

Then volume of box 

  $\begin{array}{l}V=\left(24-2x\right)\left(24-2x\right)x \\ =4\left(12-x\right)\left(12-x\right)x \\ $ =4x{(12-x)}^2$\end{array}$

b.

To determine

To fill the given table.

Explanation of Solution

Given:

X	3	3.5	3.9	4	4.1	4.5	5
V

The behaviour of the function as x approaches 4 i.e

  $\underset{x\to 4}{\lim }$ V , where $V=4x{(12-x)}^2$

X	3	3.5	3.9	4	4.1	4.5	5
V	972	1011.5	1023.5	1024	1023.5	1012.5	980

  $\begin{array}{l}\text{If }x\text{=3 then }v\text{ is 972}\\ $ \text{=}\text{4}x{\text{(12-}x\text{)}}^{\text{2}}$\text{=}{\text{4(3)(12-3)}}^{\text{2}}\\ \text{=}{\text{(12)(9)}}^{\text{2}}\\ \text{=}\text{972}\\ \\ \text{If }x\text{=3}\text{.5 then }v\text{ is 1011}\text{.5}\\ $ \text{=}\text{4}x{\text{(12-}x\text{)}}^{\text{2}}$\text{=}\text{4(3}\text{.5)(12-3}{\text{.5)}}^{\text{2}}\\ \text{=}\text{(14)(8}{\text{.5)}}^{\text{2}}\\ \text{=}\text{1011}\text{.5}\\ \\ \text{If }x\text{=3}\text{.9 then }v\text{ is 1023}\text{.5}\\ $ \text{=}\text{4}x{\text{(12-}x\text{)}}^{\text{2}}$\text{=}\text{4(3}\text{.9)(12-3}{\text{.9)}}^{\text{2}}\\ \text{=}\text{(15}\text{.6)(8}{\text{.1)}}^{\text{2}}\\ \text{=}\text{1023}\text{.5}\\ \\ \text{If }x\text{=4 then }v\text{ is 4 }\\ $ \text{=}\text{4}x{\text{(12-}x\text{)}}^{\text{2}}$\text{=}{\text{4(4)(12-4)}}^{\text{2}}\\ \text{=}{\text{(16)(8)}}^{\text{2}}\\ \text{=}\text{1023}\text{.5}\\ \\ \text{If }x\text{=4}\text{.1 then }v\text{ is1023}\text{.5}\\ $ \text{=}\text{4}x{\text{(12-}x\text{)}}^{\text{2}}$\text{=}\text{4(4}\text{.1)(12-4}{\text{.1)}}^{\text{2}}\\ \text{=}\text{(16}\text{.4)(7}{\text{.9)}}^{\text{2}}\\ \text{=}\text{1023}\text{.5}\\ \\ \text{If }x\text{=4}\text{.5 then }v\text{ is 1012}\text{.5}\\ $ \text{=}\text{4}x{\text{(12-}x\text{)}}^{\text{2}}$\text{=}\text{4(4}\text{.5)(12-4}{\text{.5)}}^{\text{2}}\\ \text{=}\text{(18)(7}{\text{.5)}}^{\text{2}}\\ \text{=}\text{1012}\text{.5}\\ \\ \text{If }x\text{=5 then }v\text{ is 980}\\ $ \text{=}\text{4}x{\text{(12-}x\text{)}}^{\text{2}}$\text{=4(4}{\text{.5)(12-5)}}^{\text{2}}\\ {\text{=(18)(7)}}^{\text{2}}\\ \text{=980}\end{array}$

c.

To determine

To plot the graph.

Explanation of Solution

Given :

X	3	3.5	3.9	4	4.1	4.5	5
V	972	1011.5	1023.5	1024	1023.5	1012.5	980

Graph :

  

Interpretation :

After inspecting this table, it conclude that the functional values less than 4 appear to be decreasing whereas the functional values greater than 4 is also appear to be decreasing.In this graph all the points are overlapping to each other as they are nearest to 1000.