On a calm, bright day, you're scuba diving in a deep lake (n = 1.33), 6.30 m away from your sailboat in the horizontal direction. When you are 2.70 m below the surface, the top of the sailboat mast appears to you to be in a direction 31.0° from vertical. Calculate the height of the mast
To calculate the height of the mast, we can use the concept of refraction and trigonometry.
Let's assume the height of the mast is "h" meters.
Given:
Distance from the sailboat to the scuba diver (horizontal distance) = 6.30 m
Depth of the scuba diver below the surface = 2.70 m
Refractive index of water = 1.33
Angle of apparent direction of the mast from vertical = 31.0°
Now, we can use the concept of refraction to set up a right triangle with the height of the mast as one of the sides.
Step 1: Calculate the distance between the sailboat mast and the scuba diver underwater.
Using trigonometry, we have:
distance = horizontal distance / cos(angle)
distance = 6.30 m / cos(31.0°)
distance ≈ 7.234 m
Step 2: Calculate the apparent depth of the mast.
Using the concept of refraction and the refractive index of water, we have:
apparent depth = depth of the scuba diver × refractive index of water
apparent depth = 2.70 m × 1.33
apparent depth ≈ 3.591 m
Step 3: Set up a right triangle to calculate the height of the mast.
The right triangle consists of the following sides:
- The vertical side representing the height of the mast (h).
- The horizontal side representing the distance underwater (apparent depth).
- The hypotenuse representing the total distance between the sailboat mast and the scuba diver (distance).
Using the Pythagorean theorem, we have:
distance^2 = apparent depth^2 + h^2
(7.234 m)^2 = (3.591 m)^2 + h^2
h^2 = (7.234 m)^2 - (3.591 m)^2
h^2 ≈ 33.542 m^2
h ≈ √33.542 m
h ≈ 5.795 m
Therefore, the height of the mast is approximately 5.795 meters.
To calculate the height of the mast, we can utilize the principles of refraction. Refraction occurs when light passes from one medium to another, such as from water to air, and causes a change in the direction of light rays.
In this scenario, we have a source of light (the top of the mast) that is submerged in water and observed by someone scuba diving. The observation is made at an angle from the vertical, which suggests that refraction is taking place.
To begin solving this problem, we can use Snell's law, which relates the angles of incidence and refraction to the refractive indices of the two media involved. Snell's law is expressed as:
n1 * sinθ1 = n2 * sinθ2
Where:
- n1 is the refractive index of the medium in which the light ray is coming from
- θ1 is the angle of incidence (in this case, the observed angle from the vertical)
- n2 is the refractive index of the medium in which the light ray is entering
- θ2 is the angle of refraction
In our case, the light rays are passing from water (n = 1.33) to air (n = 1.00). We know the value of n1 (1.33) and θ1 (31.0°), and we want to find the height of the mast, which corresponds to the vertical distance from the scuba diver to the top of the mast.
To find θ2, we rearrange Snell's law as follows:
sinθ2 = (n1 / n2) * sinθ1
θ2 = arcsin((n1 / n2) * sinθ1)
Applying the values, we get:
θ2 = arcsin((1.33 / 1.00) * sin(31.0°))
By evaluating the above expression, we find that θ2 is approximately 39.37°.
Now we have two angles: θ1 (the observed angle) and θ2 (the angle of refraction). These angles help us determine the height of the mast.
Considering the triangle formed by the scuba diver, the top of the mast, and the point on the surface directly above the mast, we can use the tangent function to relate the height of the mast (h) to the known values:
tanθ1 = h / d
tanθ2 = (h + 2.70) / D
where:
- θ1 is the observed angle from the vertical (31.0°)
- θ2 is the angle of refraction (39.37°)
- h is the height of the mast
- d is the horizontal distance between the scuba diver and the mast (6.30 m)
- D is the distance between the scuba diver and the point directly above the mast (unknown)
By rearranging the equations, we can solve for h and D simultaneously:
h = d * tanθ
D = (h + 2.70) / tanθ2
Plugging in the known values, we get:
h = 6.30 m * tan(31.0°)
D = (h + 2.70) / tan(39.37°)
Evaluating these expressions, we find that h is approximately 3.64 m and D is approximately 10.38 m.
Therefore, the height of the mast is approximately 3.64 meters.