1. Given that points S and R on opposite sides of a lake, triangle SRT is formed. To find the distance RS across the lake, a surveyor lays off RT = 53.1 m, with angle T = 32deg 10' and anagle S = 57deg 50'. Find length RS.
RS/sin T = RT/sin S
now just plug in your numbers.
To find the length of RS, we can use the Law of Cosines. The Law of Cosines is a formula that relates the lengths of the sides of a triangle to the cosine of one of its angles.
In this case, we have a triangle SRT with side lengths SR and RT, and angles S and T. We want to find the length of RS, which is opposite angle R.
The Law of Cosines states that for a triangle with side lengths a, b, and c, and angle C opposite side c, the following equation holds:
c^2 = a^2 + b^2 - 2ab * cos(C)
In our case, we have RT = 53.1 m, angle T = 32 degrees 10 minutes (or 32.167 degrees), and angle S = 57 degrees 50 minutes (or 57.833 degrees).
Now, let's calculate the length of RS using the Law of Cosines.
c^2 = a^2 + b^2 - 2ab * cos(C)
RS^2 = SR^2 + RT^2 - 2 * SR * RT * cos(T)
We need to solve for SR, so rearrange the equation:
SR^2 = RS^2 + RT^2 - 2 * RS * RT * cos(T)
Now, substitute the given values:
SR^2 = RS^2 + (53.1 m)^2 - 2 * RS * (53.1 m) * cos(32.167 degrees)
This equation can be solved to find the value of SR. Square root both sides to solve for SR:
SR = sqrt(RS^2 + (53.1 m)^2 - 2 * RS * (53.1 m) * cos(32.167 degrees))
Now, you can substitute the measurements of the angles and calculate the value of SR.