We are given a right triangle $KLM$ with a right angle at $L$. We are given that side $LM$ has length 72 and angle $M$ is 22 degrees. We are asked to find the length of side $KM$, labeled as $x$, and round to the nearest tenth.

1. Problem Description

We are given a right triangle $KLM$ with a right angle at $L$. We are given that side $LM$ has length 72 and angle $M$ is 22 degrees. We are asked to find the length of side $KM$, labeled as $x$, and round to the nearest tenth.

2. Solution Steps

Since we have a right triangle, we can use trigonometric ratios. We are given the length of the side adjacent to angle $M$ and we want to find the length of the hypotenuse. Therefore, we can use the cosine function.

$$\cos(M) = \frac{\text{adjacent}}{\text{hypotenuse}}$$

$$\cos(22^\circ) = \frac{72}{x}$$

To solve for $x$, we can multiply both sides by $x$ and then divide by $\cos(22^\circ)$:

$$x \cos(22^\circ) = 72$$

$$x = \frac{72}{\cos(22^\circ)}$$

Using a calculator, we find that $\cos(22^\circ) \approx 0.92718$.

$$x = \frac{72}{0.92718} \approx 77.65$$

Rounding to the nearest tenth, we get $x \approx 77.7$.

3. Final Answer

$x = 77.7$