Final answer:
Using the Law of Sines with the given angles of 57° and 73°, and side AB of 24 inches, the length of AC in the oblique triangle ABC is determined to be approximately 21.924 inches.
Step-by-step explanation:
To determine the length of AC in the oblique triangle ABC, we need to use the Law of Sines, as we have two angles and one side (AAS case). The Law of Sines states that the ratio of the length of a side to the sine of its opposite angle is constant for all sides and angles in a given triangle. In triangle ABC, given angle A equals 57°, angle B equals 73°, and side AB equals 24 inches, we can find angle C by subtracting the known angles from 180°, because the sum of angles in a triangle is always 180 degrees.
Angle C = 180° - 57° - 73° = 50°. Now applying the Law of Sines, we get the following proportions: (AB/sin(A)) = (AC/sin(C)). We can solve for the length of AC using the sine of angle C and AB:
AC = AB × (sin(C) / sin(A))
= 24 in × (sin(50°) / sin(57°))
= 24 in × (0.7660 / 0.8387)
= 24 in × 0.9135
= 21.924 inches (rounded to three decimal places)
So, the length of AC is approximately 21.924 inches.