Answer:
D. 13.2
Step-by-step explanation:
The law of sines tells you side lengths are proportional to the sines of their opposite angles.
Estimate
From the law of sines, we have ...
b/a = sin(B)/sin(A) = sin(49°)/sin(16°)
You know the sine function is concave downward, so the sine of 49° cannot be more than 49/16 = 3 1/16 times the sine of 16°. That means ...
b/a < 3 1/16 ⇒ b < (4)(3 1/16) = 12 1/4
The triangle inequality tells you ...
c < a +b
c < 4 + 12 1/4 = 16 1/4
Of the possible answer choices, the only sensible choice is 13.2.
Working out
If you actually want to work the problem in detail, you need to find angle C:
C = 180° -A -B
C = 180° -16° -49° = 115°
Then the law of sines tells you ...
c/a = sin(C)/sin(A)
c = a·sin(C)/sin(A) = 4·sin(115°)/sin(16°)
c ≈ 13.2
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Additional comment
We can revisit the estimating process using the actual value of angle C.
C = 180° -16° -49° = 115°
sin(C) = sin(115°) = sin(65°) . . . . . . sine identity
Again, ...
c/a < C/A
c < a(C/A) = 4(65°/16°) = 16 1/4 . . . . same as above.
