object 380 ft above surface of globe. Horizon is 65.3 degrees from object.
What is circumference of globe?
To determine the circumference of the globe, we need to use the information given about the object's height above the surface of the globe and the angle to the horizon.
First, let's visualize the situation. Imagine a right-angled triangle where the object is at the top of the triangle, the center of the globe is at the bottom vertex, and the line connecting the object and the center of the globe represents the radius of the globe. The line from the object to the horizon forms the hypotenuse of the triangle.
We can use trigonometry to find the length of the hypotenuse (radius of the globe) and then calculate the circumference.
Using the information given, we have:
The height of the object above the surface of the globe = 380 ft
Angle to the horizon = 65.3 degrees
Let's label the sides of the triangle:
Opposite Side (height of the object) = H = 380 ft
Hypotenuse (radius of the globe) = R (unknown)
Angle to the horizon = θ = 65.3 degrees (converted to radians for calculations)
To find the length of the hypotenuse, we can use the trigonometric function sine:
sin(θ) = H / R
Rearranging the formula to solve for R:
R = H / sin(θ)
Substituting the given values:
R = 380 ft / sin(65.3 degrees)
Now, we can use a calculator to find the length of R.
R ≈ 422.48 ft
Once we have the radius of the globe, we can calculate the circumference using the formula:
Circumference = 2πR
Substituting the value of R:
Circumference = 2π × 422.48 ft
Now, we can evaluate this expression using a calculator to find the approximate circumference of the globe based on the given information.