To solve for [tex]\( x \)[/tex] in the equation
[tex]\[
\cos \left(20^{\circ}\right) = \frac{4}{x},
\][/tex]
we will follow these steps:
1. Identify the given cosine value: We need the cosine of 20 degrees. Using a calculator or trigonometric tables, we find:
[tex]\[
\cos(20^\circ) \approx 0.9397.
\][/tex]
2. Set up the equation: Substitute the known cosine value into the equation:
[tex]\[
0.9397 = \frac{4}{x}.
\][/tex]
3. Solve for [tex]\( x \)[/tex]: We need to rearrange the equation to solve for [tex]\( x \)[/tex].
Multiply both sides of the equation by [tex]\( x \)[/tex]:
[tex]\[
0.9397x = 4.
\][/tex]
Next, divide both sides by 0.9397 to isolate [tex]\( x \)[/tex]:
[tex]\[
x = \frac{4}{0.9397}.
\][/tex]
4. Compute the value of [tex]\( x \)[/tex]:
[tex]\[
x \approx \frac{4}{0.9397} \approx 4.2567.
\][/tex]
5. Round to the nearest hundredth: Finally, round the result to the nearest hundredth:
[tex]\[
x \approx 4.26.
\][/tex]
Thus, the value of [tex]\( x \)[/tex] rounded to the nearest hundredth is
[tex]\[
\boxed{4.26}.
\][/tex]
