A vertical pole stands on a 20° slope from 50 m down the slope from the pole the angle of elevation of the top of the pole is 35 degrees. How tall is the pole

angle of 25 degrees is measured, point 'M'

our obtuse triangle is then 'TBM' and we also define a point 'G' that is at ground level and colinear with points 'T' and 'B' such that 'MBG' forms a right triangle whose hypotenuse is 240 feet
we want to solve for angle BMG, the angle the side of the hill makes with the horizontal
since triangle MBG is a right triangle, once we solve for angle MBG, we know BMG since the sum of the angles in any triangle is 180 degrees... and we know that the supplement to angle MBG is angle MBT, therefore once we solve for angle MBT we can easily calculate the desired angle, BMG
let 'x' represent angle MTB, then by the law of sines, we can solve for angle MTB:
sin(x)/240 = sin(25)/125
x = asin((sin(25)/125)*240) = 54.235
we can solve for angle MBT since the sum of the angles in any triangle is 180 degrees:
180 - 25 - 54.235 = 100.77
as stated, angle MBG is supplemental to angle MBT:
angle MBG = 180 - 100.77 = 79.23
finally, angle GMB is based on the sum of the angles in a triangle equal to 180 degrees:
180 - 90 - 79.23 = 10.77
the angle the side of the hill makes with the horizontal is approximately 10.8 degrees
A tower 125 feet high stands on the side of a hill. At a point 240 feet from the foot of the tower measured straight down the hill, the tower subtends an angle of 25 degrees. What angle does the side of the hill make with the horizontal?
• math - Steve, Friday, January 3, 2014 at 5:50pm
Draw a diagram.
T = top of tower
B = base of tower
D = point 240' downhill

To find the height of the pole, we can use trigonometry. Let's call the height of the pole "h".

First, let's draw a diagram to visualize the situation.

/|
/ |
/ | h
/ |
/_______________|
50m 20° | 35°

In this right triangle, the angle of elevation is the angle between the horizontal ground and the line of sight to the top of the pole. The angle of depression is the angle between the horizontal ground and the line of sight from the top of the pole to a point farther down the slope.

We can see that the angle of depression from the top of the pole to the point 50m down the slope is 20°.

Using trigonometry, we can set up the following equation:

tan(20°) = (h + 50m) / h

Now we can solve for h:

tan(20°) = (h + 50) / h

To solve for h, we can simplify the equation by multiplying both sides by h:

h * tan(20°) = h + 50

Next, let's isolate h by subtracting h from both sides of the equation:

h * tan(20°) - h = 50

Factor out h from the left side of the equation:

h * (tan(20°) - 1) = 50

Finally, divide both sides of the equation by (tan(20°) - 1):

h = 50 / (tan(20°) - 1)

Using a calculator, we can find the value of the expression (tan(20°) - 1), which is approximately 0.364.

Therefore, the height of the pole, h, is given by:

h = 50 / 0.364

Calculating this, we find that h is approximately 137.36 meters.

Thus, the height of the pole is approximately 137.36 meters.