Final answer:
The longer diagonal of a rhombus with sides of 14 inches and obtuse angles of 160 degrees is approximately 27.7 inches, calculated using trigonometric functions and rounding to three significant figures.
Step-by-step explanation:
To find the length of the longer diagonal of a rhombus with side length 14 inches and one obtuse angle measuring 160 degrees, we need to apply trigonometric functions. Since opposite angles of a rhombus are equal, the other obtuse angle is also 160 degrees, meaning each acute angle is 180 - 160 = 20 degrees. The diagonals of a rhombus bisect each other at right angles and also bisect the angles of the rhombus.
Let d be the length of the longer diagonal, and p the length of the shorter diagonal, which is divided into two equal segments by the bisecting diagonal. Therefore, we get two 10 degrees angles. Applying the sine function to one of these right-angled triangles formed by the diagonals and one side of the rhombus, we can write: sin(10 degrees) = (p/2) / 14. Solving for p/2 gives us p/2 = 14 * sin(10 degrees).
To find the length of d, we use the cosine function with the angle adjacent to the longer diagonal: cos(10 degrees) = (d/2) / 14. Solving for d/2 yields d/2 = 14 * cos(10 degrees). Therefore, d is equal to 2 * 14 * cos(10 degrees).
After calculating the value, we get the length of the longer diagonal d ≈ 27.7 inches (rounded to three significant figures).