Answer:
(a) 16.7 units
Explanation:
You want the length of the side opposite the angle 68° in a triangle with a side of length 18 opposite the angle 86°.
Law of sines
The law of sines tells you side lengths are proportional to the sine of the opposite angle:
AC/sin(B) = BC/sin(A)
AC = BC·sin(B)/sin(A)
Angle B is a little more than 3/4 of angle A, so the ratio of sines will be more than that value, but less than 1. This tells you AC < (3/4)BC, eliminating choices b, c, d.
The length of AC is about 16.7 units.
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Additional comment
If you put the numbers into the expression for AC and do the math, you find AC ≈ 16.7301° ≈ 16.7, as we estimated.
68/86 ≈ 0.7907
sin(68)/sin(86) ≈ 0.9294
The ratio of sines of angles versus the angle ratio is only a good match for small angles (generally 5° or less). Otherwise, the ratio of the smallest to largest angle will always be less than the ratio of their sines. (This is because the sine function has decreasing slope for first-quadrant angles.)
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