Find the height of the dam given angles of a triangle The top of a dam has an angle of elevation of 1.3 radians from a point on a river. Measuring the angle of elevation to the top of the dam from a point 155 feet farther downriver is 0.8 radians; assume the two angle measurements are taken at the same elevation above sea level. How high is the dam? So I made two equations: y= height 1) tan (0.8)= y/(155+a) 2) tan (1.3)= y/a So then finding the tan of those and creating y=..... and got: 1)y=0.0139635a+2.1643489 2) y= 0.0222693a I set those two equal to eachother and I round to the nearest foot and keep getting 248 ft but it is wrong. I feel like Im doing this problem right, but somehow it's wrong Here's the pic: !enter image description here

The tangent of the angle of elevation is the ratio of the dam height $h$ to the horizontal distance downriver of the observer. This gives us two equations for the height:

$$h=a\tan{(1.3)}=(155+a)\tan{(0.8)}$$

$$\implies a=\frac{h}{\tan{(1.3)}},~~~155+a=\frac{h}{\tan{(0.8)}}. $$

Subtracting the first equation from the second,

$$155=(155+a)-a=\frac{h}{\tan{(0.8)}}-\frac{h}{\tan{(1.3)}}=h\left(\frac{1}{\tan{(0.8)}}-\frac{1}{\tan{(1.3)}}\right).$$

Solving for $h$,

$$h=155 \text{ ft }\left(\frac{1}{\tan{(0.8)}}-\frac{1}{\tan{(1.3)}}\right)^{-1}\\\ \approx 223\text{ ft }.$$