Final answer:
To find the length of n, we can use the law of sines and solve an equation involving angles L, M, and N. By plugging in the given values, we can find the length of n to the nearest inch.
Step-by-step explanation:
To find the length of n in triangle LMN, we need to use the law of sines. The law of sines states that the ratio of the sine of an angle to the length of the side opposite that angle is constant for all angles in a triangle. Using this law, we can set up the following equation:
(sin N) / n = (sin L) / l
Plugging in the given values, we have:
(sin N) / n = (sin 40°) / 170
Multiplying both sides by n gives:
sin N = (n / 170) * sin 40°
Next, we can use the inverse sine function to find the value of N:
N = arcsin((n / 170) * sin 40°)
Finally, we can solve for n by plugging in the given value of 121° for angle M and using the fact that angle L + angle M + angle N = 180°:
180 - 121 - N = L
180 - 121 - arcsin((n / 170) * sin 40°) = 170
Solving this equation will give the length of n to the nearest inch.