A spherical generator is used to produce a -15.00D surface on glass of refractive index 1.80. The diameter of the cutting tool is 80mm and the radius of the cutting surface is 4mm. What is the angle between the axis of the tool and the axis of the lens?

To find the angle between the axis of the tool and the axis of the lens, we can use trigonometry.

Let's first understand the given information:
1. The diameter of the cutting tool is 80mm, which means the radius of the cutting tool is 40mm.
2. The radius of the cutting surface is 4mm.

To visualize the situation, imagine a cross-section of the lens and cutting tool:

L -> Lens
T -> Cutting Tool

_______T______ __________
|\ r_c \ \ /
| \ \ \ /
| \ \ \ /
| \ \____\L/
| \
| \
|_____________|

To find the angle between the axes, we need to calculate the difference in height (h) between the axis of the lens and the axis of the cutting tool.

The height (h) can be found using the formula:

h = r_c - r_t

Where:
r_c is the radius of the cutting surface (4mm)
r_t is the radius of the cutting tool (40mm)

h = 4mm - 40mm = -36mm

Now we can use the height (h) to find the angle (θ) between the axes.

tan(θ) = h / length of lens

The length of the lens can be calculated using the formula:

length of lens = diameter of cutting tool / (2 * index of refraction)

Given that the diameter of the cutting tool is 80mm and the refractive index is 1.80, we can calculate the length of the lens as follows:

length of lens = 80mm / (2 * 1.80) = 22.22mm

Now we can substitute the values into the equation for tangent:

tan(θ) = (-36mm) / 22.22mm

Using a scientific calculator or table, we can find the inverse tangent (arctan) of this ratio to find the angle (θ) between the axes. The calculated angle will be in radians, so we can convert it to degrees if needed.

θ ≈ -59.02 degrees

Therefore, the angle between the axis of the tool and the axis of the lens is approximately -59.02 degrees.