The question gives us a right-angled triangle and find the value of x.
In order to solve the problem, we use SOHCAHTOA. In this case, we will use "SOH" from SOHCAHTOA because we have the Opposite as x and Hypotenuse as 24, while the relevant angle is 60 degrees.
Let us apply this formula:
[tex]\begin{gathered} \text{ SOH implies:} \\ \sin \theta=\frac{\text{Opposite}}{\text{Hypotenuse}} \\ \\ \theta=60^0,\text{Opposite}=x,\text{Hypotenuse}=24 \\ \\ \therefore\sin 60^0=\frac{x}{24} \end{gathered}[/tex]We simply need to make x the subject of the formula and we shall also represent sin 60 with its surd form.
This is done below:
[tex]\begin{gathered} \sin 60^0=\frac{x}{24} \\ \text{ Multiply both sides by 24} \\ 24\times\sin 60^0=\frac{x}{24}\times24 \\ \therefore x=24\times\sin 60^0 \\ \\ \sin 60^0=\frac{\sqrt[]{3}}{2} \\ \\ x=24\times\frac{\sqrt[]{3}}{2}=12\times2\times\frac{\sqrt[]{3}}{2}\text{ (2 crosses out)} \\ \\ x=12\sqrt[]{3} \end{gathered}[/tex]Therefore, the final answer is Option 4