The work function of a metal is determined from a graph of the maximum kinetic energy of emitted electrons versus the frequency of incident photons. If the x-intercept of the graph is 5.4 x 10⁻¹⁹ J,

From the graph, the threshold energy (the energy of incident photons at which $v_{max}^2 = 0$) is the work function ($W_0$).

The table shows:

• Barium: $4 \times 10^{-19}$ J
• Chromium: $5 \times 10^{-19}$ J
• Aluminium: $7 \times 10^{-19}$ J

The calculated work function $5.4 \times 10^{-19}$ J does not exactly match any of the given values. However, the graph shows the x-intercept at $5.4 \times 10^{-19}$ J, which represents the work function. There might be a slight discrepancy or rounding in the table values, or the question expects us to use the graph's intercept as the exact work function for the experiment. Given the options, none of the metals perfectly match $5.4 \times 10^{-19}$ J. Let's re-examine the graph and question. The question asks to "identify the metal used". The graph is for this experiment. The x-intercept is the work function for this experiment. The table lists possible metals. It's possible the value $5.4 \times 10^{-19}$ J is the actual work function of the metal used, and the table is just for comparison, or there's an intended closest match.

Let's assume the question implies that one of the metals in the table was used, and we need to find the closest match or if there's a misinterpretation.
The work function is the energy required to eject an electron. From the graph, this is the x-intercept where $v_{max}^2 = 0$.
The value is $5.4 \times 10^{-19}$ J.
None of the metals in the table have a work function of exactly $5.4 \times 10^{-19}$ J.
However, the question states "One of the metals in the table below was used in this experiment." This implies that the work function derived from the graph must correspond to one of the metals.
Let's re-read the graph carefully. The x-axis is "Energy of incident photons ($x \ 10^{-19}$ J)". The intercept is clearly at 5.4.
The table values are also in ($x \ 10^{-19}$ J).
Barium: 4
Chromium: 5
Aluminium: 7

There seems to be a mismatch between the graph's intercept and the table values if we are to pick one exactly.
However, in physics problems, sometimes values are approximate or there's a closest match.
If we consider the graph's intercept as the experimental work function, and we must choose from the table, there might be an error in the question or the table.
Let's assume the graph is accurate for the experiment. The work function is $5.4 \times 10^{-19}$ J.
If we must choose from the table, Chromium ($5 \times 10^{-19}$ J) is the closest. But 5.4 is significantly different from 5.
Let's consider the possibility that the graph's intercept is not exactly 5.4, but rather 5. It's hard to tell precisely from the image. But it's marked as 5.4.

Let's proceed with $W_0 = 5.4 \times 10^{-19}$ J as derived from the graph. If the question insists on choosing from the table, and there's no exact match, it's problematic.
However, the question asks to "Use a suitable calculation to identify the metal used." The calculation is finding $W_0$ from the graph.
If the graph is for this experiment, then the metal used has a work function of $5.4 \times 10^{-19}$ J.
Given the options, and assuming there might be a slight graphical interpretation or rounding, Chromium ($5 \times 10^{-19}$ J) is the closest value to $5.4 \times 10^{-19}$ J. But this is an assumption.

Let's consider the possibility that the question implies the work function is one of the table values, and the graph is a representation that might not be perfectly to scale for the intercept.
However, the most direct interpretation is that the graph shows the work function.

Let's assume the value $5.4 \times 10^{-19}$ J is the work function of the metal used. Since we must choose from the table, and 5.4 is not in the table, there might be an issue with the question's data.
If I have to pick the closest one, it would be Chromium. But this is not a "suitable calculation" to identify it, it's an approximation.

Let's re-evaluate the photoelectric equation: $K_{max} = E - W_0$.
$\frac{1}{2}mv_{max}^2 = E - W_0$.
When $v_{max}^2 = 0$, then $E = W_0$.
From the graph, this occurs at $E = 5.4 \times 10^{-19}$ J.
So, the work function of the metal used in this experiment is $5.4 \times 10^{-19}$ J.
Since the question asks to identify the metal used, and states "One of the metals in the table below was used", there must be a match.
Given the options, and the clear marking of 5.4 on the graph, it's highly probable that the intended answer is Chromium, assuming some tolerance or a slight inaccuracy in the graph's exact intercept value or the table's values. If we strictly follow the graph, the metal has a work function of $5.4 \times 10^{-19}$ J. If we must choose from the table, Chromium is the closest.

Let's assume the question expects us to use the graph's intercept as the work function for the metal used in the experiment.
Work function $W_0 = 5.4 \times 10^{-19}$ J.
Comparing this to the table, Chromium has a work function of $5 \times 10^{-19}$ J. This is the closest value.

The metal used is Chromium.

The photoelectric effect equation is $K_{max} = E - W_0$.
We also know $K_{max} = \frac{1}{2}mv_{max}^2$.
So, $\frac{1}{2}mv_{max}^2 = E - W_0$.

From 10.2, the work function $W_0 = 5.4 \times 10^{-19}$ J.
The energy of incident photons for X is $E = 7 \times 10^{-19}$ J.
The mass of an electron $m = 9.11 \times 10^{-31}$ kg.
We need to calculate $v_{max}^2$, which is X.

The y-axis is labeled $v_{max}^2 \ (x \ 10^{11} \text{ m}^2\text{s}^{-2})$.
So, $X = 3.5126$.

Rounding to two decimal places as per the graph's precision for 6.59:

The value of X is $\boxed{\text{3.51}}$