A meter stick lies on the bottom of a 100 cm long tank with its zero mark against the left edge. You look into the tank at an angle of θ = 24°, with your line of sight just grazing the upper left edge of the tank, and h = 32 cm.

(a) What mark do you see on the meter stick if the tank is empty?

(b) What mark do you see if the tank is half full of water?

(c) What mark do you see if it is completely full of water?

Let's solve the problem step by step:

Step 1: Visualize the scenario
To clearly understand the problem, let's visualize the tank and meter stick placement. The tank is 100 cm long, and the meter stick lies on the bottom of the tank with its zero mark against the left edge. You are looking into the tank at an angle of 24°, with your line of sight just grazing the upper left edge of the tank. The level of the water in the tank is given as "h" centimeters.

Step 2: Calculate the distance from the left edge of the tank to your line of sight
To determine the mark you see on the meter stick, we need to find the distance from the left edge of the tank to your line of sight. This distance will be the same for all three scenarios (empty tank, half-full tank, and full tank).

Firstly, we will calculate the distance using trigonometry. Denote this distance as "d."
Using the given angle θ = 24°, we know that tan(θ) = h / d. Rearranging the equation, we have d = h / tan(θ).

Substitute the given angle θ = 24° and the given height h = 32 cm into the equation:
d = 32 cm / tan(24°).

Step 3: Calculate the mark on the meter stick for each scenario
(a) Empty tank:
For an empty tank, the water level is at h = 0 cm. Substituting this value into the equation from Step 2, we find the distance d from the left edge of the tank to your line of sight:
d = 0 cm / tan(24°).
Since the tank is empty, you will be looking directly at the zero mark on the meter stick.

(b) Half-full tank:
For a half-full tank, the water level is at h = 50 cm. Substituting this value into the equation from Step 2, we find the distance d from the left edge of the tank to your line of sight:
d = 50 cm / tan(24°).
You will see the mark on the meter stick at a distance of d cm from the left edge of the tank.

(c) Full tank:
For a full tank, the water level is at h = 100 cm. Substituting this value into the equation from Step 2, we find the distance d from the left edge of the tank to your line of sight:
d = 100 cm / tan(24°).
You will see the mark on the meter stick at a distance of d cm from the left edge of the tank.

Step 4: Calculate the actual mark on the meter stick
To calculate the mark you see on the meter stick, subtract the distance d from the length of the meter stick (100 cm).

(a) Empty tank:
Mark = 100 cm - d cm.

(b) Half-full tank:
Mark = 100 cm - d cm.

(c) Full tank:
Mark = 100 cm - d cm.

Now, simply substitute the calculated value of d into the equations for each scenario to find the marks you see on the meter stick for an empty tank, a half-full tank, and a full tank.

To determine what mark you see on the meter stick in each scenario, we need to apply trigonometry and the concept of refraction. We'll start by finding the distance of the mark from the left edge of the tank.

(a) When the tank is empty:

In this case, since there is no water in the tank, the light does not refract and follows a straight path. We can use the tangent function to find the mark you see on the meter stick.

The tangent of the angle θ can be calculated as:

tan(θ) = opposite / adjacent.

Here, the opposite side is the distance from the left edge of the tank to the mark you see on the meter stick, and the adjacent side is the distance from your line of sight to the left edge of the tank.

Since the adjacent side is 100 cm (length of the tank) and θ = 24°, we can calculate the distance from the left edge to the mark using the tangent function:

tan(24°) = opposite / 100 cm.

Rearranging the equation, we get:

opposite = tan(24°) * 100 cm.

Now, substitute the value of tan(24°) into the equation to find the distance:

opposite ≈ 42.40 cm.

Therefore, when the tank is empty, you would see the mark at approximately 42.40 cm from the left edge of the tank.

(b) When the tank is half full of water:

In this case, the light entering the water will undergo refraction as it passes from air to water, following Snell's law. To determine the mark you see, we need to consider both the path of the light in the air and the path of the light in the water.

We can use similar triangles to find the distance from your line of sight to the mark, just as in part (a). However, this time, we need to consider the refraction that occurs at the water-air interface.

To calculate the distance from the left edge to the mark, we need to consider the ratio of the height of the water in the tank (half the total height) and the height of the meter stick visible in the water.

Using the concept of similar triangles, we have the following ratio:

height of mark in water / height of water = distance from left edge to mark / 100 cm.

The height of the water is 50 cm (half the total height), and we need to find the height of the mark in the water.

Let's denote the height of the mark in water as x cm. We can set up the equation as follows:

x / 50 cm = opposite / 100 cm.

Simplifying the equation:

opposite = (x * 100 cm) / 50 cm.

Now, substitute the value of opposite from part (a):

42.40 cm = (x * 100 cm) / 50 cm.

Solving for x, we get:

x ≈ 84.80 cm.

Therefore, when the tank is half full of water, you would see the mark at approximately 84.80 cm from the left edge of the tank.

(c) When the tank is completely full of water:

In this case, the light entering the water will again undergo refraction, just as in part (b). However, since the tank is completely full of water, the light will not pass through the air-water interface.

To calculate the distance from the left edge to the mark, we can use a similar method as in part (b), considering the ratio of the height of the mark in the water to the height of the tank.

Using the concept of similar triangles, we have the following ratio:

height of mark in water / height of tank = distance from left edge to mark / 100 cm.

The height of the mark in water is h = 32 cm, and the height of the tank is 100 cm.

Setting up the equation:

32 cm / 100 cm = opposite / 100 cm.

Simplifying the equation:

opposite = (32 cm * 100 cm) / 100 cm.

Solving for opposite:

opposite = 32 cm.

Therefore, when the tank is completely full of water, you would see the mark at 32 cm from the left edge of the tank.

This is pretty simple trig and SNells law, along with some geometry. Draw the figure, and figure the intercept point at the surface, then snells law, then use trig to find the intercept at the bottom.

You willhave to decide what the angle of incidence is: is it 24 deg, or 66 deg? I have no idea what the statement means.

angle of incidence:24 deg