Find $$ c $$ if $$ a=2.51 \mathrm{mi}, b=3.66 \mathrm{mi} $$ and $$ \angle C=41.5 $$ degrees. Enter $$ c $$ rounded to 3 decimal places. $$ c=\square \mathrm{mi} $$; Assume $$ \angle A $$ is opposite side $$ a, \angle B $$ is opposite side $$ b $$, and $$ \angle C $$ is opposite side

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To find the length of side $$ c $$ in a triangle given sides $$ a $$ and $$ b $$ and the included angle $$ C $$, we can use the Law of Cosines. The Law of Cosines states:

$$
c^2 = a^2 + b^2 - 2ab \cos(C)
$$

Given:
- $$ a = 2.51 $$ mi
- $$ b = 3.66 $$ mi
- $$ C = 41.5^\circ $$

First, we need to calculate $$ \cos(41.5^\circ) $$. Using a calculator:

$$
\cos(41.5^\circ) \approx 0.7547
$$

Now we can substitute the values into the Law of Cosines formula:

$$
c^2 = (2.51)^2 + (3.66)^2 - 2 \cdot (2.51) \cdot (3.66) \cdot \cos(41.5^\circ)
$$

Calculating each term:

1. $$ (2.51)^2 = 6.3001 $$
2. $$ (3.66)^2 = 13.3956 $$
3. $$ 2 \cdot (2.51) \cdot (3.66) \cdot \cos(41.5^\circ) = 2 \cdot 2.51 \cdot 3.66 \cdot 0.7547 $$

Calculating the last term:

$$
2 \cdot 2.51 \cdot 3.66 \cdot 0.7547 \approx 2 \cdot 2.51 \cdot 3.66 \cdot 0.7547 \approx 2 \cdot 9.1866 \cdot 0.7547 \approx 13.8511
$$

Now substituting back into the equation:

$$
c^2 = 6.3001 + 13.3956 - 13.8511
$$

Calculating the right side:

$$
c^2 = 6.3001 + 13.3956 - 13.8511 \approx 5.8446
$$

Now, taking the square root to find $$ c $$:

$$
c = \sqrt{5.8446} \approx 2.415
$$

Finally, rounding to three decimal places:

$$
c \approx 2.415 \text{ mi}
$$

Thus, the value of $$ c $$ is:

$$
\boxed{2.415} \text{ mi}
$$
$$ c \approx 2.415 $$ mi

Find if and degrees.
Enter rounded to 3 decimal places.
;
Assume is opposite side is opposite side , and is opposite side

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Find if and degrees. Enter rounded to 3 decimal places. ; Assume is opposite side is opposite side , and is opposite side