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. The formula to calculate the apparent depth of an object in a different medium is given by:
Apparent depth = Actual depth / Refractive index
In this case, the refractive index is given as n = 1.33.
We can use this formula to calculate the apparent depth of the sailboat mast from the scuba diver's perspective.
The actual depth of the mast is the sum of the depth of the scuba diver (2.70 m) and the distance from the scuba diver to the mast (6.30 m), which is 9.00 m.
Apparent depth = 9.00 m / 1.33 = 6.77 m
Now, we can use trigonometry to calculate the height of the mast.
Given that the angle between the line of sight from the scuba diver to the top of the mast and the vertical direction is 31.0°, we can use the tangent function:
tan(31.0°) = Height of mast / Apparent depth
Height of mast = tan(31.0°) × Apparent depth
Height of mast = tan(31.0°) × 6.77 m
Using a calculator, we can find the height of the mast to be approximately 3.64 m.
Therefore, the height of the mast is approximately 3.64 meters.
To calculate the height of the mast, we can use the concept of refraction. The apparent height of the mast is different due to the bending of light as it passes from water (n = 1.33) to air (n = 1.00). Here's how you can solve it step by step:
1. Determine the angle of refraction: The angle of refraction can be calculated using Snell's Law, which states that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the ratio of the refractive indices of the two mediums. In this case, the angle of incidence is 31.0°, and the refractive indices are 1.33 (water) and 1.00 (air). Using Snell's Law: n1 * sin(theta1) = n2 * sin(theta2), where n1 and n2 are the refractive indices of the two mediums, and theta1 and theta2 are the angles of incidence and refraction, respectively. Rearranging the equation, we get sin(theta2) = (n1 / n2) * sin(theta1). Plugging in the values: sin(theta2) = (1.33 / 1.00) * sin(31.0°).
2. Calculate the true height of the mast: To find the true height, we need to determine the vertical distance between your eye level and the top of the mast. Let's call this distance 'h'. Using basic trigonometry, h = 2.70 m * tan(theta2).
3. Find the apparent height of the mast: The apparent height is the height of the mast as observed by you underwater. Let's call this height 'h_apparent'. Using basic trigonometry, h_apparent = (6.30 m * tan(31.0°)) + h.
4. Calculate the height of the mast: The height of the mast is the difference between the apparent height and the vertical distance between your eye level and the surface of the water. Let's call this distance 'd'. d = 2.70 m + (6.30 m * tan(31.0°)). Finally, the height of the mast is h_mast = h_apparent - d.
By following these steps, you'll be able to calculate the height of the mast. Plug in the values and perform the calculations to get the final result.