The peak of a mountain is observed from the base and from the top of a tower 180 ft high. Find the height of the mountain above the base of the tower is the angles of elevation of the peak are 21 degree 35’ and 24 degree 48’
To find the height of the mountain above the base of the tower, we can use trigonometry and set up a right triangle.
Let's label the height of the mountain as "h" and the distance between the base of the tower and the mountain as "x".
From the base of the tower, the angle of elevation to the peak of the mountain is 21 degrees 35 minutes. We can convert this angle to decimal degrees by dividing the minutes by 60 and adding it to the degrees:
21 + 35/60 = 21.5833 degrees
Similarly, for the top of the tower, the angle of elevation to the peak of the mountain is 24 degrees 48 minutes. Converting this angle to decimal degrees:
24 + 48/60 = 24.8000 degrees
Now, let's consider the right triangle formed by the base of the tower, the top of the tower, and the peak of the mountain. The height of the tower is given as 180 ft.
Using trigonometry, we can set up the following relationship:
tan(angle) = opposite / adjacent
For the first angle (21.5833 degrees):
tan(21.5833 degrees) = h / x
For the second angle (24.8000 degrees):
tan(24.8000 degrees) = (h + 180) / x
Now we have two equations with two variables (h and x). We can solve this system of equations to find the values of h and x.
First, divide both sides of the first equation by x:
tan(21.5833 degrees) = h / x
h / x = tan(21.5833 degrees)
Then, divide both sides of the second equation by x:
tan(24.8000 degrees) = (h + 180) / x
(h + 180) / x = tan(24.8000 degrees)
Now we can solve the system of equations using algebraic methods. Multiply both sides of the first equation by x and rearrange:
h = x * tan(21.5833 degrees)
Substitute this value for h into the second equation:
(x * tan(21.5833 degrees) + 180) / x = tan(24.8000 degrees)
Next, cross multiply:
x * tan(24.8000 degrees) = x * tan(21.5833 degrees) + 180
Expand and rearrange:
x * tan(24.8000 degrees) - x * tan(21.5833 degrees) = 180
Factor out x:
x * (tan(24.8000 degrees) - tan(21.5833 degrees)) = 180
Finally, solve for x by dividing both sides of the equation by (tan(24.8000 degrees) - tan(21.5833 degrees)):
x = 180 / (tan(24.8000 degrees) - tan(21.5833 degrees))
Once we have the value of x, we can substitute it back into the first equation to find the height of the mountain:
h = x * tan(21.5833 degrees)
After calculating these equations, the value of h will give us the height of the mountain above the base of the tower.
As usual, draw a diagram. It is clear that if the height of the mountain is h, and the tower is at a distance x from the mountain,
(h-180)/x = tan 21°35'
h/x = tan 24°48'
Eliminate x and you have
(h-180)/tan21°35' = h/tan24°48'
Now just solve for h