Fortune tellers use crystal balls to see the future. A fortune teller has a crystal ball with an index of refraction of 1.5 and diameter of 0.2 m. You sit on one side of the ball and the fortune teller sits on the other. She holds up a small red jewel on her side in the equatorial plane of the ball. Let P be the point on the ball nearest to the jewel. How far away in m from P is the image of the jewel on your side?

Hint: the question mark in the image below is an unknown you'll need to figure out as an intermediate step. However, the requested answer is the distance from P, i.e. ? + 0.2 m.

Details and assumptions
The jewel is at a distance of 0.8 m from P.

To solve this problem, we can use the concept of refraction and the relationship between object distance, image distance, and the index of refraction. Here's how we can approach it:

Step 1: Identify the given information:
- diameter of the crystal ball: 0.2 m
- index of refraction of the crystal ball: 1.5
- distance between the jewel and point P: 0.8 m

Step 2: Draw a diagram:
To visualize the situation, draw a diagram with the crystal ball, the jewel, the point P, and a ray of light passing through the ball from the jewel:

Fortune Teller | You
|
P |
__________ | __________
/ | \
| | |
Jewel | Imaginary
| Jewel
ball |
| |
| |

Step 3: Identify the relevant formulas:
For refraction through a spherical surface, we can use the formula known as Snell's law:

n1*sin(theta1) = n2*sin(theta2)

Where,
n1 = index of refraction of the medium the light ray is coming from (in this case, air)
theta1 = angle of incidence (measured with respect to the normal to the surface)
n2 = index of refraction of the medium the light ray is entering (in this case, the crystal ball)
theta2 = angle of refraction (measured with respect to the normal to the surface)

The angles of incidence and refraction are related to the object and image distances as follows:

Object distance / Image distance = tan(theta1) / tan(theta2)

Step 4: Calculate the angle of incidence:
The distance between the jewel and P is given as 0.8 m. Since P is the nearest point on the ball, it lies on the radius. So, the radius of the ball is, R = 0.2 / 2 = 0.1 m.

We can calculate the angle of incidence using the object distance and the radius of the ball:

tan(theta1) = object distance / radius = 0.8 m / 0.1 m = 8

Taking the inverse tangent of both sides, we get:

theta1 = tan^(-1)(8) ≈ 82.81 degrees

Step 5: Calculate the angle of refraction:
We know the index of refraction of the crystal ball is 1.5. Plugging in the values, we can use Snell's law to find the angle of refraction.

n1 * sin(theta1) = n2 * sin(theta2)
(1) * sin(82.81 degrees) = (1.5) * sin(theta2)

sin(theta2) = sin(82.81 degrees) / 1.5
theta2 = sin^(-1)(sin(82.81 degrees) / 1.5) ≈ 56.54 degrees

Step 6: Calculate the image distance:
To find the image distance, we can rearrange the ratio of object distance to image distance:

Object distance / Image distance = tan(theta1) / tan(theta2)

Since the ratio of diameters in similar triangles is equal to the ratio of distances, we have:

Image distance = object distance * (tan(theta2) / tan(theta1))
= 0.8 m * (tan(56.54 degrees) / tan(82.81 degrees))
≈ 0.8 m * (1.619 / 8.165)
≈ 0.1592 m

Step 7: Calculate the distance from P:
The requested answer is the distance from P, which is the image distance plus the diameter of the crystal ball:

Distance from P = Image distance + Diameter of the ball
= 0.1592 m + 0.2 m
= 0.3592 m

Therefore, the distance from P to the image of the jewel on your side is approximately 0.3592 meters.