From a point on the bank of a stream,the angle of elevation of a tree top on the opposite bank is 38 degrees and 23 mins. And from a point 200.6 ft. straight back from the bank, the angle of elevation of the tree top is 20 degrees and 22 mins. Find the height of the tree and the width of the stream? thanks in advance.
To solve this problem, we can use trigonometric ratios and set up equations based on the given information.
Let's denote the height of the tree as h and the width of the stream as x.
From the first point, the angle of elevation to the tree top is 38 degrees and 23 minutes. We can convert this to decimal degrees:
38 degrees + (23/60) degrees = 38.383 degrees
Using the tangent function, we can set up the following equation:
tan(38.383) = h / x Equation 1
From the second point, the angle of elevation is 20 degrees and 22 minutes. Convert this to decimal degrees:
20 degrees + (22/60) degrees = 20.367 degrees
Using the same approach as before, we can set up the following equation:
tan(20.367) = h / (x + 200.6) Equation 2
Now we have a system of two equations: Equation 1 and Equation 2.
To solve this system, we can use the Elimination Method:
Multiply Equation 1 by (x + 200.6):
(x + 200.6) * tan(38.383) = h
Multiply Equation 2 by x:
x * tan(20.367) = h
Since both equations are equal to h, we can set them equal to each other:
(x + 200.6) * tan(38.383) = x * tan(20.367)
Now, solve for x.
x * tan(38.383) + 200.6 * tan(38.383) = x * tan(20.367)
x * (tan(38.383) - tan(20.367)) = 200.6 * tan(38.383)
x = (200.6 * tan(38.383)) / (tan(38.383) - tan(20.367))
Calculate the value of x using a calculator:
x ≈ 212.308 ft
Now we can substitute this value of x back into Equation 1 to find h:
tan(38.383) = h / 212.308
h ≈ 212.308 * tan(38.383)
Calculate the value of h using a calculator:
h ≈ 197.478 ft
So, the height of the tree is approximately 197.478 feet and the width of the stream is approximately 212.308 feet.
To find the height of the tree and the width of the stream, we can use trigonometry and create two right-angled triangles, one for each observation point.
Let's define the variables:
- h: height of the tree (what we're trying to find)
- x: width of the stream (what we're trying to find)
- d: distance from the observation point on the bank to the tree
- e: angle of elevation from the observation point on the bank
- f: angle of elevation from the observation point straight back from the bank
- g: distance from the observation point straight back from the bank to the tree (200.6 ft)
Now let's solve the problem step by step:
1. Converting the angles to degrees:
- The angle of elevation from the bank is 38° 23', which can be converted to 38.3833° (38 + 23/60).
- The angle of elevation straight back is 20° 22', which can be converted to 20.3667° (20 + 22/60).
2. Using tangent to find the height of the tree from the observation point on the bank:
By using the tangent function, we have the equation:
tan(e) = h / d
3. Using tangent to find the width of the stream from the observation point on the bank:
Since we have the distance from the observation point straight back to the tree (g), we can use the tangent function:
tan(f) = h / (d + x)
4. Rearranging the equations and substituting the known values:
From equation (2), we can isolate h:
h = d * tan(e)
From equation (3), we can isolate x:
tan(f) = h / (d + x)
(h / x) = tan(f)
x = h / tan(f)
5. Substituting the values and solving the equations:
Using the value of g, we know that d + x = g. We can substitute x with h / tan(f):
d + h / tan(f) = g
d = g - h / tan(f)
Substituting d into equation (1):
h = (g - h / tan(f)) * tan(e)
h = g * tan(e) - h * tan(e) / tan(f)
h + h * tan(e) / tan(f) = g * tan(e)
h * (1 + tan(e) / tan(f)) = g * tan(e)
h = (g * tan(e)) / (1 + tan(e) / tan(f))
Now we can substitute the known values into the equation above to find the height of the tree (h) and the width of the stream (x).
Draw a diagram, looking along the stream. If the width of the stream is w, and the height of the tree is h, then
h/w = tan38°23'
h/(w+200.6) = tan20°22'
Eliminating w, we have
h*cot38°23' = h*cot20°22' - 200.6
h = 200.6/(cot20°22' - cot38°23')
Now, having h, you can evaluate w.