Points M and N are separated by an obstacle. In order to find the distance between them, a third point P is slected which is 120 yards from M and 150 yeards from N. The angle MPN is measured to be 80deg10'. Find the distance from M to N.
To find the distance between points M and N, we can use the Law of Cosines. The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles.
Let's label the distance between M and N as "d". To apply the Law of Cosines, we need to find the length of the side opposite the angle MPN, which is the side connecting points P and N. We'll label this side as "a", which has a length of 150 yards. The length of the side connecting points P and M, which we'll label as "b", is 120 yards. Finally, the measure of angle MPN is given as 80 degrees 10 minutes.
The Law of Cosines states that:
c^2 = a^2 + b^2 - 2ab * cos(C)
where c represents the unknown side (in this case, the distance between M and N), a and b are the known lengths of the other two sides, and C is the measure of the angle opposite the unknown side.
Now, let's plug in the values into the formula:
d^2 = 150^2 + 120^2 - 2 * 150 * 120 * cos(80deg10')
To calculate the cosine of the angle in degrees, we need to convert the angle from degrees and minutes to decimal form. In this case, 80 degrees 10 minutes is equal to 80.167 degrees.
d^2 = 22500 + 14400 - 2 * 150 * 120 * cos(80.167)
Now, we need to calculate the cosine of 80.167 degrees. You can use a scientific calculator or an online calculator that allows angles to be entered in degrees. The cosine of 80.167 degrees is approximately 0.173.
d^2 = 22500 + 14400 - 2 * 150 * 120 * 0.173
Evaluate the expression on the right side of the equation:
d^2 = 22500 + 14400 - 49800
d^2 = -12900
Since it doesn't make sense for the squared distance to be negative, there must be an error in the given information or calculations. Please double-check the values and calculations to ensure accuracy.