A submarine is travelling parallel to the surface of the ocean at a depth of 626 m. It begins a constant ascent
in order to reach the surface after travelling a distance of 4420 m. The ascent takes 35 min.
a) What angle of ascent, or angle of elevation, would the submarine need to make to reach the surface in
4420 m? Answer to the nearest degree.
b) How far did the submarine travel horizontally during its ascent to the surface? Answer to the nearest
metre.
a) We can use the tangent function to find the angle of ascent:
tan(angle) = opposite/adjacent
In this case, the opposite side is the vertical distance the submarine needs to travel to reach the surface, which is 626 m. The adjacent side is the horizontal distance the submarine travels during the ascent, which is 4420 m.
tan(angle) = 626/4420
angle = tan^-1(626/4420)
Using a calculator, we get angle ≈ 8.1 degrees when rounded to the nearest degree.
b) We can use trigonometry again to find the horizontal distance travelled during the ascent. The horizontal distance is the adjacent side of the angle we just found, which is 4420 m.
cos(angle) = adjacent/hypotenuse
The hypotenuse is the distance travelled by the submarine during the ascent, which we can find by multiplying the ascent rate (distance/time) by the ascent time:
hypotenuse = (4420/35) * 60 = 7554.3 m
cos(angle) = 4420/7554.3
horizontal distance = adjacent = cos(angle) * hypotenuse
Using a calculator, we get horizontal distance ≈ 2586 m when rounded to the nearest metre.
a) To find the angle of ascent, we can use the tangent function. The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the depth of the submarine (626 m) and the adjacent side is the distance it travels horizontally (4420 m).
Let's calculate the angle:
Tangent of angle = opposite / adjacent
Tangent of angle = 626 / 4420
Now, let's find the angle by taking the inverse tangent (arctan) of the ratio:
Angle = arctan(626 / 4420)
Angle ≈ 7.99
So, the angle of ascent is approximately 8 degrees.
b) To find the horizontal distance traveled during the ascent, we can use the sine function. The sine of an angle is equal to the opposite side divided by the hypotenuse. In this case, the opposite side is the horizontal distance traveled and the hypotenuse is the total distance traveled (4420 m).
Let's calculate the horizontal distance:
Sine of angle = opposite / hypotenuse
Sine of angle = horizontal distance / 4420
Now, let's find the horizontal distance by rearranging the equation:
Horizontal distance = Sine of angle * 4420
Using the angle from part (a), we can calculate the horizontal distance:
Horizontal distance = Sin(8 degrees) * 4420
Horizontal distance ≈ 636 m
Therefore, the submarine traveled approximately 636 meters horizontally during its ascent to the surface.