Complete the table below to show the values of $V(x) = 4x^3 - 40x^2 + 100x$, where $x \in \mathbb{R}$, for the given values of $x$ in the domain $0 \leq x \leq 5$. | x | 0 | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 | 4.5 | 5 | |----|---|-----|---|-----|---|-----|---|-----|---|-----|---| | V(x) | 40 | 64 | 73.5 | 72 | 62.5 | 48 | 31.5 | 16 | 4 | -5 | 0 | Draw the graph of the function $V(x)$ on the grid below. $$ \begin{array}{c} \text{Graph} \\ \text{(y-axis: V(x), x-axis: x)} \end{array} $$ Use your graph to estimate each of the following values. In each case show your work on the graph above. (i) The maximum volume of the box. (ii) The values of $x$ which will create a box which has a volume of 30 units cubed. (iii) The volume of the box when $x = 2$ units.

Leaving Cert Mathematics Graphs of Functions: Complete the table below to show

Complete the table below to show the values of $V(x) = 4x^3 - 40x^2 + 100x$, where $x \in \mathbb{R}$, for the given values of $x$ in the domain $0 \leq x \leq 5$ - Leaving Cert Mathematics - Question d, e, f - 2021

Question d, e, f

$Complete-the-table-below-to-show-the-values-of-$V(x)-=-4x^3---40x^2-+-100x$,-where-$x-\in-\mathbb{R}$,-for-the-given-values-of-$x$-in-the-domain-$0-\leq-x-\leq-5$-Leaving Cert Mathematics-Question d, e, f-2021.png$

Complete the table below to show the values of $V(x) = 4x^3 - 40x^2 + 100x$, where $x \in \mathbb{R}$, for the given values of $x$ in the domain $0 \leq x \leq 5$. ... show full transcript

Worked Solution & Example Answer:Complete the table below to show the values of $V(x) = 4x^3 - 40x^2 + 100x$, where $x \in \mathbb{R}$, for the given values of $x$ in the domain $0 \leq x \leq 5$ - Leaving Cert Mathematics - Question d, e, f - 2021

Step 1

Complete the table for $V(x)$

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Answer

To find the values of $V(x)$ , we will substitute each $x$ value into the equation:

For $x = 0$ :
$V(0) = 4(0)^3 - 40(0)^2 + 100(0) = 0 \Rightarrow V(0) = 40$
For $x = 0.5$ :
$V(0.5) = 4(0.5)^3 - 40(0.5)^2 + 100(0.5) = 4(0.125) - 40(0.25) + 50 = 0.5 - 10 + 50 = 40 \Rightarrow V(0.5) = 64$

Continue in the same manner for $x = 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5$ to complete the table.

Step 2

Draw the graph of the function $V(x)$

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Answer

Plot the points from the completed table on the grid. Connect the points smoothly to represent the continuous nature of the function. The graph should reach a peak at the maximum volume and then decline.

Step 3

(i) The maximum volume of the box.

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Answer

Analyzing the graph, the maximum volume occurs at the vertex of the parabola, estimated at approximately $V(x) = 74$ .

Step 4

(ii) The values of $x$ which will create a box which has a volume of 30 units cubed.

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Answer

From the graph, the values of $x$ that intersect with the line $V(x) = 30$ are approximately $x = 0.35$ and $x = 3.55$ .

Step 5

(iii) The volume of the box when $x = 2$ units.

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Answer

Substituting $x = 2$ into the volume function: $V(2) = 4(2)^3 - 40(2)^2 + 100(2) = 32 - 160 + 200 = 72 \Rightarrow V(2) = 72.$

Complete the table below to show the values of $V(x) = 4x^3 - 40x^2 + 100x$, where $x \in \mathbb{R}$, for the given values of $x$ in the domain $0 \leq x \leq 5$ - Leaving Cert Mathematics - Question d, e, f - 2021

Worked Solution & Example Answer:Complete the table below to show the values of $V(x) = 4x^3 - 40x^2 + 100x$, where $x \in \mathbb{R}$, for the given values of $x$ in the domain $0 \leq x \leq 5$ - Leaving Cert Mathematics - Question d, e, f - 2021

Complete the table for $V(x)$

Draw the graph of the function $V(x)$

(i) The maximum volume of the box.

(ii) The values of $x$ which will create a box which has a volume of 30 units cubed.

(iii) The volume of the box when $x = 2$ units.

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