Right Triangle Applications

[nycmathguy]

College Algebra
Chapter 1/Section 3

Radius of Earth = 3960 miles
Feet in one mile = 5280

For 53

Let d = hypotenuse

(d)^2 + (radius of Earth)^2 = [(radius of Earth) plus (distance above sea level)/(feet in one mile)]^2

(d)^2 + (3960)^2 = [(3960) + (20/5280)]^2

Is this the correct set up?

Can you set up 54? I will show the math work?

 

[MathLover1]

53. set up is correct

54.


Use the facts that the radius of Earth is 3960 miles and 1 mile =5280 feet.

Solution. The Earth is a sphere, so we cut this sphere with a plane
passing through the person, the ship, and the center of the Earth. This
gives the following cross section (certainly not to scale!):

The radius of the Earth in feet is (3960 miles)(5280 feet/mile) = (3960)(5280) feet = 20,908,800 feet.
So the distance from the center of the Earth to the eyes of the observer (technically, to the top of the observer’s head ) is 20,908,806 feet.
We draw a line tangent to the circle passing through the observer’s eyes. Lines tangent to a circle are perpendicular to a radius of the circle containing the point of tangency, so we get the pictured right triangle and we want to find d.

Since we have a right triangle, then the Pythagorean Theorem gives
(20,908,800 ft)^2 + d^2 = (20,908,806 ft)^2
d^2 = (20,908,806^2 − 20,908,800)^2ft^2
d = sqrt(250905636ft^2 )

Since d is a distance it is positive and so
d =15840 ft
Now (15840 feet feet)(1/5280miles/feet) = 3 miles.

So the ship is 3 miles away .

 

[nycmathguy]

53. set up is correct

54.


Use the facts that the radius of Earth is 3960 miles and 1 mile =5280 feet.

Solution. The Earth is a sphere, so we cut this sphere with a plane
passing through the person, the ship, and the center of the Earth. This
gives the following cross section (certainly not to scale!):
View attachment 3824
The radius of the Earth in feet is (3960 miles)(5280 feet/mile) = (3960)(5280) feet = 20,908,800 feet.
So the distance from the center of the Earth to the eyes of the observer (technically, to the top of the observer’s head ) is 20,908,806 feet.
We draw a line tangent to the circle passing through the observer’s eyes. Lines tangent to a circle are perpendicular to a radius of the circle containing the point of tangency, so we get the pictured right triangle and we want to find d.

Since we have a right triangle, then the Pythagorean Theorem gives
(20,908,800 ft)^2 + d^2 = (20,908,806 ft)^2
d^2 = (20,908,806^2 − 20,908,800)^2ft^2
d = sqrt(250905636ft^2 )

Since d is a distance it is positive and so
d =15840 ft
Now (15840 feet feet)(1/5280miles/feet) = 3 miles.

So the ship is 3 miles away .

A perfect, detailed reply for study notes.