The problem asks to solve for $x$ in the right triangle $LMN$. We are given that angle $N$ is $64^\circ$, side $LN$ is $45$, and side $LM$ is $x$. We need to round the final answer to the nearest tenth.

1. Problem Description

The problem asks to solve for $x$ in the right triangle $LMN$. We are given that angle $N$ is $64^\circ$, side $LN$ is $45$, and side $LM$ is $x$. We need to round the final answer to the nearest tenth.

2. Solution Steps

Since triangle $LMN$ is a right triangle with right angle at vertex $M$, we can use trigonometric ratios to relate the sides and angles. We have the following information:

* Angle $N = 64^\circ$

* Side $LN$ (adjacent to angle $M$) = $45$

* Side $LM$ (opposite to angle $N$) = $x$

We can use the tangent function:

$tan(N) = \frac{opposite}{adjacent}$

$tan(64^\circ) = \frac{x}{45}$

$x = 45 * tan(64^\circ)$

Using a calculator:

$tan(64^\circ) \approx 2.0503$

$x \approx 45 * 2.0503$

$x \approx 92.2635$

Rounding to the nearest tenth, we get:

$x \approx 92.3$

3. Final Answer

$x \approx 92.3$