Bob is driving along a straight and level road straight toward a mountain. at some point on his trip he measures the angle of elevation to the top of the mountain and finds it to be 23 degrees 42 minutes. he then drives 1 mile more and measures the angle of elevation to be 30 degrees 38 minutes. find the height of the mountain to the nearest foot.
To find the height of the mountain, we can use trigonometry and the given angles of elevation.
Let's assume that the distance Bob drives from the first measurement point to the mountain is x miles.
Using the tangent function, we can relate the height of the mountain (h) to the angle of elevation (θ) and the distance from the observer to the object (x).
For the angle of elevation of 23 degrees 42 minutes, the tangent of the angle is:
tan(θ) = h / x
Converting the angle to decimal degrees:
θ = 23 + (42 / 60) = 23.7 degrees
Similarly, for the angle of elevation of 30 degrees 38 minutes:
θ = 30 + (38 / 60) = 30.6333 degrees
Now, we can set up two equations using the tangent function and solve for x and h:
Equation 1: tan(23.7) = h / x
Equation 2: tan(30.6333) = h / (x + 1)
Solving Equation 1 for h:
h = tan(23.7) * x
Substituting this into Equation 2:
tan(30.6333) = (tan(23.7) * x) / (x + 1)
Now we can solve for x:
tan(30.6333) * (x + 1) = tan(23.7) * x
Using a scientific calculator, we find x ≈ 1.215 miles.
Then, we can substitute this value of x back into Equation 1 to find h:
h = tan(23.7) * 1.215
Using a scientific calculator, we find h ≈ 0.48 miles.
Finally, to convert the height to feet, we multiply by 5280 feet per mile:
Height ≈ 0.48 * 5280 ≈ 2534.4 feet
Therefore, the height of the mountain is approximately 2534.4 feet to the nearest foot.
To find the height of the mountain, we can use the concept of trigonometry. Let's denote the distance Bob drove from the first measuring point to the second as "x" miles.
We can establish a right triangle between the top of the mountain, the first measuring point, and the mountain's base. The angle of elevation at the first measuring point gives us an angle; we'll call it angle A. The angle of elevation at the second measuring point gives us another angle; we'll call it angle B.
We need to find the height of the mountain, which is equivalent to the vertical distance from the first measuring point to the top of the mountain. Let's call this height "h."
Now, let's break down the problem using trigonometric functions, specifically the tangent function:
In the first right triangle:
tan(A) = h / x (Equation 1)
In the second right triangle:
tan(B) = h / (x + 1) (Equation 2)
We have two equations with two unknowns (h and x). To solve this system of equations, we can substitute equation 1 into equation 2. By doing that, we eliminate "h" as follows:
tan(B) = (tan(A) * (x + 1)) / x
Now, let's plug in the given values:
A = 23 degrees 42 minutes = 23.7 degrees
B = 30 degrees 38 minutes = 30.63 degrees
Converting to radians:
A = 23.7 * (pi / 180)
B = 30.63 * (pi / 180)
Substitute those values into the equation:
tan(30.63 * (pi / 180)) = (tan(23.7 * (pi / 180)) * (x + 1)) / x
Now, we can solve for "x":
x = ((tan(23.7 * (pi / 180)) * (x + 1)) / tan(30.63 * (pi / 180))) - 1
Using a calculator, we can solve this equation to find that x is approximately 2.71 miles.
Now, let's substitute this value of x into equation 1 to find "h":
h = tan(23.7 * (pi / 180)) * x
h = tan(23.7 * (pi / 180)) * 2.71
Using a calculator, we can find that h is approximately 1.14 miles.
To convert the height from miles to feet, we multiply by 5280 (since there are 5280 feet in a mile):
Height in feet = 1.14 * 5280
Height in feet ≈ 6019.2 feet
Therefore, the height of the mountain, rounded to the nearest foot, is approximately 6019 feet.
Let the height of the mountain (in feet) be H.
Let the initial horizontal distance from the mountain peak be X.
H/X = tan 23.7 = 0.43897
H/(X-5280) = tan 30.633 =0.59218
Solve the two equations in two unknowns (X and H)
H - 0.43897X = 0
H - 0.59218X = -3126.7