Two mountains are 1020 m and 1235 m high. Standing on the summit of the lower one I look up through an angle of elevation of 160 to see the summit of the higher one. Calculate the horizontal distance between the two mountains and the distance between the two summits.
difference in heights=215m
tan(16deg)=215/distance
solve for distance.
difference in heights = 1235-1020 = 215 m
I will assume your angle is 16° , not 160° , the latter would make no sense
so you are looking at a right-angled triangle with height of 215 and a base angle of 16°
horizontal distance between them ---- x
tan16° = 215/x
x = 215/tan16 = ....
let the hypotenuse, the distance between them, be h
sin16 = 215/h
h = 215/sin16 = ....
To solve this problem, we can use the trigonometric functions sine and cosine.
Let's start by finding the horizontal distance between the two mountains.
Step 1: Calculate the vertical distance between the two mountains.
To calculate the vertical distance, we can use the formula:
Vertical Distance = height of the taller mountain - height of the shorter mountain
Vertical Distance = 1235 m - 1020 m
Vertical Distance = 215 m
Step 2: Use the trigonometric function sine to find the horizontal distance.
The formula for calculating the horizontal distance can be expressed as:
Horizontal Distance = Vertical Distance / sin(angle of elevation)
Horizontal Distance = 215 m / sin(160°)
Horizontal Distance ≈ 501.55 m (rounded to two decimal places)
Now, let's calculate the distance between the two summits.
Step 3: Use the Pythagorean theorem to find the distance between the two summits.
The Pythagorean theorem states that the square of the hypotenuse (distance between the two summits) is equal to the sum of the squares of the other two sides (vertical distance and horizontal distance).
Distance between the two summits = √(Vertical Distance² + Horizontal Distance²)
Distance between the two summits = √(215 m² + 501.55 m²)
Distance between the two summits ≈ 544.71 m (rounded to two decimal places)
Therefore, the horizontal distance between the two mountains is approximately 501.55 meters, and the distance between the two summits is approximately 544.71 meters.
To calculate the horizontal distance between the two mountains, we can use trigonometry. Specifically, we can use the tangent function based on the angle of elevation.
Step 1: Determine the vertical distance between the two mountains.
The vertical distance between the two mountains can be found by subtracting the height of the lower mountain from the height of the higher mountain.
Vertical distance = 1235 m - 1020 m = 215 m
Step 2: Calculate the horizontal distance using the tangent function.
Using the angle of elevation (160 degrees) and the vertical distance (215 m), we can set up the equation:
tan(angle of elevation) = vertical distance / horizontal distance
Rearranging the equation to solve for the horizontal distance:
horizontal distance = vertical distance / tan(angle of elevation)
horizontal distance = 215 m / tan(160 degrees)
Since the tangent of 160 degrees is negative, we would need to use the absolute value to obtain a positive horizontal distance.
horizontal distance ≈ 215 m / |-0.36397023| ≈ 591.11 m
Therefore, the approximate horizontal distance between the two mountains is 591.11 meters.
To calculate the distance between the two summits, you can use the Pythagorean theorem, which relates the horizontal distance, the vertical distance, and the hypotenuse (distance between the two summits).
Using the vertical distance (215 m) and horizontal distance (591.11 m), we can set up the equation:
Distance between summits = √[(vertical distance)² + (horizontal distance)²]
Distance between summits ≈ √[(215 m)² + (591.11 m)²]
Distance between summits ≈ √[46225 m² + 349050.6721 m²]
Distance between summits ≈ √39527525.6721 m²
Distance between summits ≈ 6286.76 m
Therefore, the approximate distance between the two summits is 6286.76 meters.