You are building a (rectangular) pool in your backyard. It measures 30m in the north-south direction and 25 meters east-west. (Some pool!) The bottom of the pool is a slanted plane so that it's 1m deep along the southern edge and 10m deep along the northern edge. (As I said, some pool!)
To build the pool you compute the angle of depression of a line from the southeast corner to the northeast corner. It is how many
degrees?
Draw a diagram.
assuming the line runs along the bottom of the pool, The angle is x, where
tan(x) = 9/√(25^2+30^2)
Well, well, well, looks like you're diving into some fun geometry! Alright, let's calculate that angle of depression.
First, we need to find the horizontal distance between the southeast and northeast corners of the pool. Since the pool measures 30m north-south and 25m east-west, we have a lovely right triangle.
The height of the triangle is 10m (depth along the northern edge), and the base is 25m (width of the pool). Using our trusty Pythagorean theorem, we can find the hypotenuse, which is the horizontal distance we're after.
So, let's dust off our math skills and solve it: hypotenuse^2 = height^2 + base^2.
hypotenuse^2 = 10^2 + 25^2.
hypotenuse^2 = 100 + 625.
hypotenuse^2 = 725.
hypotenuse ≈ 26.9m (rounded to one decimal place).
Now that we have the horizontal distance, we can calculate the angle of depression. It's the angle between the horizontal line connecting the southeast and northeast corners and the line going down from the southeast corner.
To find this angle, we can use some trigonometry. We'll use the tangent function (tan) because we have the opposite (10m) and adjacent (26.9m) sides of the triangle.
Angle = tan^-1(opposite/adjacent).
Angle = tan^-1(10/26.9).
Angle ≈ 20.7 degrees (rounded to one decimal place).
So, the angle of depression between the southeast corner and the northeast corner of your majestic pool is approximately 20.7 degrees. Enjoy the dips and dives, my friend!
To find the angle of depression from the southeast corner to the northeast corner of the pool, we can use trigonometry.
First, let's consider a right-angled triangle formed by the line of sight, the height difference between the two corners (10m - 1m = 9m), and the horizontal distance between the two corners (30m).
We can use the tangent function to find the angle of depression:
tan(θ) = opposite/adjacent = 9m/30m = 0.3
To find the angle θ, we can use the inverse tangent function (arctan or tan^(-1)):
θ = arctan(0.3)
Using a calculator, we find that θ is approximately 16.7 degrees.
Therefore, the angle of depression from the southeast corner to the northeast corner of the pool is approximately 16.7 degrees.
To compute the angle of depression of a line from the southeast corner to the northeast corner, we can visualize it as a right triangle where the line is the hypotenuse. We know the lengths of the two legs of the triangle, which are the depth differences along the southern and northern edges of the pool.
Let's calculate the length of the hypotenuse first. We can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Using the given dimensions, the length of the north-south direction (vertical side) is 30 meters, and the east-west direction (horizontal side) is 25 meters. Therefore, the length of the hypotenuse can be calculated as follows:
Hypotenuse^2 = 30^2 + 25^2
Hypotenuse^2 = 900 + 625
Hypotenuse^2 = 1525
Taking the square root of both sides, we find:
Hypotenuse = √1525
Hypotenuse ≈ 39.13 meters
Now, we can calculate the angle of depression using trigonometry. The angle of depression can be defined as the angle between the line and a horizontal line parallel to the ground.
In this case, the opposite side of the angle is the depth difference between the northern and southern edges, which is 10 - 1 = 9 meters. The adjacent side is the horizontal distance, which is 25 meters.
Using the tangent function:
tan(angle) = opposite / adjacent
tan(angle) = 9 / 25
Taking the inverse tangent (arctan) of both sides to find the angle:
angle = arctan(9 / 25)
angle ≈ 20.53 degrees
Therefore, the angle of depression from the southeast corner to the northeast corner of the pool is approximately 20.53 degrees.