You are traveling in a car toward a certain mountain at a speed of 80km/hr. The apparent size of the mountain is .5degrees. Fifteen minutes later the same mountain has an apparent size of 1 degree. About how tall is the mountain?
I believe you must convert the degrees to radians, but I am not sure what to do after that. If someone could at least get me started in the right direction, I would greatly appreciate it. Thanks
350 meters
Oh, so you're on a mission to measure the height of a mountain! Well, buckle up because we're about to embark on a math-filled joyride!
To start, let's figure out how much the apparent size of the mountain has changed in radians. We know that the apparent size went from 0.5 degrees to 1 degree. But you're right, we need to convert those degrees to radians first.
There are 360 degrees in a circle, and 2π radians in a circle, so to convert degrees to radians, we can use the formula: radians = degrees * (π/180).
So, the initial apparent size of the mountain in radians will be:
0.5 * (π/180) = 0.5π/180 radians.
Similarly, the final apparent size of the mountain in radians will be:
1 * (π/180) = π/180 radians.
Now that we have the change in apparent size of the mountain in radians, we can use trigonometry to estimate its height.
Assuming the mountain is sufficiently far away, we can use the Small Angle Approximation. In this approximation, we consider the triangle formed by the observer, the top of the mountain, and the base of the mountain. The tangent of the angle (θ) between the observer's line of sight and the base (opposite side) of the mountain is equal to the height (h) of the mountain divided by the distance to the mountain (d).
So, we have tan(θ) = h/d, where θ is the change in apparent size of the mountain in radians.
Now, if we assume the height of the mountain is much greater than the distance to it, we can ignore the distance (the base of the triangle) and approximate the tangent as: tan(θ) ≈ θ.
Hence, we can say that h ≈ θ, where θ is the change in apparent size of the mountain in radians.
Considering that the initial apparent size of the mountain in radians is 0.5π/180 and the final apparent size of the mountain in radians is π/180, the change in apparent size will be:
θ = π/180 - 0.5π/180 = 0.5π/180 radians.
Therefore, the estimated height of the mountain (h) will be approximately:
h ≈ 0.5π/180.
Now, if you would like a numerical answer, we can calculate that for you. Could you please provide the actual numerical values for π and the conversion of 1 radian to kilometers?
To find the height of the mountain, we can use trigonometry and the concept of angular size.
First, let's convert the apparent size of the mountain from degrees to radians.
1 degree = π/180 radians
So, the initial apparent size of the mountain is 0.5° * π/180 = 0.0087 radians.
Next, let's calculate the angular speed of change in the apparent size of the mountain.
The change in time is 15 minutes, which is (15/60) hours = 0.25 hours.
The change in the apparent size of the mountain is from 0.0087 radians to 1.0 degree = 1.0 * π/180 radians.
To find the angular speed, divide the change in the apparent size of the mountain by the change in time:
Angular speed = (1.0 * π/180 radians - 0.0087 radians) / 0.25 hours
Simplifying this gives us the angular speed.
Now, we can use the angular speed and the initial apparent size of the mountain to find the distance to the mountain.
Distance = 2 * height * tan(apparent size / 2)
Since we already know the initial apparent size of the mountain and the angular speed, we can solve for the height using the formula:
Height = Distance / (2 * tan(apparent size / 2))
Substituting the values we have, we get:
Height = (80 km/hr * 0.25 hr) / (2 * tan(0.0087 / 2))
Now, we can calculate the height of the mountain using this equation.
To determine the height of the mountain, we can use trigonometry. We need to calculate the distance between the observer (you in the car) and the mountain at two different times, along with the change in the apparent size of the mountain.
First, let's convert the degrees to radians. Since there are π radians in a circle (360 degrees), we can use the formula: radians = degrees × (π/180).
The initial apparent size of the mountain is 0.5 degrees. Converting this to radians, we have: initial radians = 0.5 × (π/180).
Next, we need to find the change in the apparent size of the mountain. The change is given as 1 degree.
Now, we have all the necessary information to calculate the distance to the mountain at the two different times.
Let's assume that the distance to the mountain at the initial time is 'd1' and the distance at the later time is 'd2'. Using the formula for angular size:
Angular size = (actual size of the object) / (distance to the object)
We can rearrange the formula to find the distance:
Distance = (actual size of the object) / (angular size)
At the initial time, the angular size is given as 0.5 degrees (or in radians, initial radians). So, we can write:
d1 = (actual height of the mountain) / initial radians
Similarly, at the later time, the angular size is 1 degree, and we have:
d2 = (actual height of the mountain) / (1 degree × (π/180))
Now, we have two equations:
d1 = (actual height of the mountain) / initial radians
d2 = (actual height of the mountain) / (1 degree × (π/180))
We can divide these equations to eliminate the actual height of the mountain:
d1 / d2 = initial radians / (1 degree × (π/180))
Now, we can substitute the given values and find the ratio of the distances (d1/d2). The actual height of the mountain can be calculated by multiplying this ratio with the distance travelled by the car in 15 minutes (15 minutes × 80 km/hr = ? km).
So, the height of the mountain can be estimated using the formula:
Actual height of the mountain = (ratio of distances) × (distance travelled by the car)
I hope this guide helps you in solving the problem!
These are small angles, so the tangent of the small angle is the same as the angle in radians.
angle= angleindegrees(PI radians/180 deg)
= .5 (3.14/180)
Draw the system. The original distance is d
h/d=.5(3.14/180)
The next distance is 20km less, so
h/(d-20)=1.0 (3.14/180)
solve each equation for d
d= 2*180/3.14 h and
d= 180/3.14 h + 20
set these two equal, and solve for h.
check my thinking.