We have a right triangle LMN, where angle M is 90 degrees, side LN has length 45, angle N is 64 degrees, and side LM has length $x$. We need to find the value of $x$.

1. Problem Description

We have a right triangle LMN, where angle M is 90 degrees, side LN has length 45, angle N is 64 degrees, and side LM has length $x$. We need to find the value of $x$.

2. Solution Steps

Since we have a right triangle, we can use trigonometric functions. We are given the length of the hypotenuse (LN) and the measure of angle N. We want to find the length of the side opposite to angle N, which is LM ($x$). We can use the sine function to relate the angle N, the opposite side (LM), and the hypotenuse (LN):

$sin(N) = \frac{opposite}{hypotenuse}$

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

To solve for $x$, we multiply both sides of the equation by 45:

$x = 45 \cdot sin(64^{\circ})$

Using a calculator, we find that $sin(64^{\circ}) \approx 0.8988$

$x = 45 \cdot 0.8988$

$x \approx 40.446$

Rounding to the nearest tenth, we get $x \approx 40.4$

3. Final Answer

$x \approx 40.4$