solve for (x). round to the nearest tenth, if necessary.

Explanation:

Step1: Identify the trigonometric ratio

In right - triangle $MNO$ with right - angle at $N$, we know the side adjacent to angle $M$ ($MN = 6.3$) and we want to find the hypotenuse $x$ (side $MO$). We use the cosine function since $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Here, $\theta = 75^{\circ}$ and the adjacent side to $\theta$ is $6.3$. So, $\cos(75^{\circ})=\frac{6.3}{x}$.

Step2: Solve for $x$

We can rewrite the equation $\cos(75^{\circ})=\frac{6.3}{x}$ as $x=\frac{6.3}{\cos(75^{\circ})}$. We know that $\cos(75^{\circ})=\cos(45^{\circ}+ 30^{\circ})=\cos45^{\circ}\cos30^{\circ}-\sin45^{\circ}\sin30^{\circ}=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}-\frac{\sqrt{2}}{2}\times\frac{1}{2}=\frac{\sqrt{6}-\sqrt{2}}{4}\approx0.259$. Then $x = \frac{6.3}{0.259}\approx24.3$.

Answer:

$24.3$