To find the value of [tex]\( n \)[/tex] to the nearest whole number, we will apply the Law of Cosines, which states:
[tex]\[
a^2 = b^2 + c^2 - 2 b c \cos(A)
\][/tex]
Given the values:
- [tex]\( a = 22 \)[/tex]
- [tex]\( b = 29 \)[/tex]
- [tex]\( c = 41 \)[/tex]
- [tex]\( A = 18^\circ \)[/tex]
First, we need to convert the angle [tex]\( A \)[/tex] from degrees to radians, because the cosine function typically works with radians in mathematical calculations. The conversion from degrees to radians is done using the formula:
[tex]\[
\text{radians} = \text{degrees} \times \frac{\pi}{180}
\][/tex]
Therefore, [tex]\( A = 18^\circ \)[/tex] converted to radians is:
[tex]\[
A = 18 \times \frac{\pi}{180} = \frac{\pi}{10} \approx 0.314 \text{ radians}
\][/tex]
Applying the Law of Cosines:
[tex]\[
a^2 = b^2 + c^2 - 2 b c \cos(A)
\][/tex]
Plugging in the given values:
[tex]\[
22^2 = 29^2 + 41^2 - 2 \cdot 29 \cdot 41 \cos(0.314)
\][/tex]
Calculating the squares:
[tex]\[
484 = 841 + 1681 - 2 \cdot 29 \cdot 41 \cos(0.314)
\][/tex]
Adding the squares:
[tex]\[
484 = 2522 - 2 \cdot 29 \cdot 41 \cos(0.314)
\][/tex]
Now, compute the cosine part. Let's call this value [tex]\( x \)[/tex]:
[tex]\[
x = 2 \cdot 29 \cdot 41 \cos(0.314)
\][/tex]
[tex]\[
x \approx 2 \cdot 29 \cdot 41 \cdot 0.951 \approx 2263.262
\][/tex]
Continuing with the equation:
[tex]\[
484 = 2522 - 2263.262
\][/tex]
Solving this step-by-step:
[tex]\[
484 = 258.738
\][/tex]
To find the value of [tex]\( n \)[/tex], we solve for [tex]\( n \)[/tex]:
[tex]\[
n^2 = 258.738
\][/tex]
Taking the square root of both sides:
[tex]\[
n = \sqrt{258.738} \approx 16.09
\][/tex]
Therefore, rounding [tex]\( n \)[/tex] to the nearest whole number gives:
[tex]\[
n \approx 16
\][/tex]
Thus, the value of [tex]\( n \)[/tex] to the nearest whole number is:
[tex]\[
\boxed{16}
\][/tex]
