Question

If in an oblique triangle ABC, side a = 17 ft, side b = 20 ft, and angle C = 19 degrees, then what is the length of side c to the nearest foot?

Answer 1

To find the length of side \(c\) in triangle \(ABC\) using the Law of Cosines, we can use the following formula:

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

Given:
- \(a = 17\) ft
- \(b = 20\) ft
- \(C = 19^\circ\)

First, we compute \(a^2\) and \(b^2\):

\[
a^2 = 17^2 = 289
\]
\[
b^2 = 20^2 = 400
\]

Next, we calculate \(2ab\):

\[
2ab = 2 \times 17 \times 20 = 680
\]

Now, we need to find \(\cos(19^\circ)\). Using a calculator, we find:

\[
\cos(19^\circ) \approx 0.9455
\]

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

\[
c^2 = 289 + 400 - 680 \times 0.9455
\]

Calculating \(680 \times 0.9455\):

\[
680 \times 0.9455 \approx 642.34
\]

So, we can update our equation:

\[
c^2 = 289 + 400 - 642.34
\]
\[
c^2 = 689 - 642.34
\]
\[
c^2 = 46.66
\]

Now, taking the square root to find \(c\):

\[
c = \sqrt{46.66} \approx 6.83
\]

Rounding to the nearest foot, we find:

\[
c \approx 7 \text{ ft}
\]

Thus, the length of side \(c\) is approximately:

\[
\boxed{7} \text{ ft}
\]