Question

Several small drops of paint (less than \(1 \mathrm{~mm}\) in diameter) splatter on a painter's eyeglasses, which are approximately \(2 \mathrm{~cm}\) in front of the painter's eyes. Do the dots appear in what the painter sees? How do the dots affect what the painter sees?

Short answer

Answer: Yes, the small paint drops on the painter's eyeglasses can affect their vision. The paint drops have an angular size of approximately 2.86°, which is significant enough to be noticeable and may obstruct a portion of the painter's field of view, potentially affecting their ability to paint precisely.

Step-by-step solution

Convert the paint drop size and the distance from the eye to meters

First, we need to convert the size of the paint drops and the distance from the eyes to the same unit, which is meters (m). The paint drops are less than 1mm in diameter, and eyeglasses are about 2cm in front of the painter's eyes. We can convert these values as follows: Paint drop diameter: \(1 mm = 0.001 m\) Distance from the eyes: \(2 cm = 0.02 m\)

Calculate the angular size of the paint drops

Next, we need to find the angular size (θ) of the paint drops as seen by the painter. We can use the formula for the small angle approximation, where: \( θ = \frac{d}{D}\), where d is the diameter of the paint drops and D is the distance from the eyes.

Plug in values and perform the calculation

Now we can plug in the values we obtained in step 1: \( θ = \frac{0.001}{0.02}\) \( θ = 0.05\) radian To convert the angle from radians to degrees, we can use the following formula: \( θ_{degrees} = \frac{θ_{radians}}{\pi} × 180\) \( θ_{degrees} = \frac{0.05}{\pi} × 180\) \( θ_{degrees} ≈ 2.86°\)

Interpret the results

The angular size of the paint drops is approximately 2.86°. Although the paint drops are small and close to the eyes, their angular size is significant enough to be noticed by the painter. This means that the small drops of paint will indeed appear in what the painter sees, and they will affect their vision. The drops can obstruct a portion of the painter's field of view, creating a nuisance and potentially affecting their ability to paint precisely. In summary, the small paint drops on the painter's eyeglasses will be noticeable to the painter and may affect their vision by obstructing a portion of their field of view.