Shadow and Penumbra
1. A 1.9 m tall person is standing next to a building. The shadow of the building projected by sunlight is 90 m while the shadow of the person is 9 m. How tall is the building?
We begin the problem by thinking about the sun's rays, since they must be parallel to each other. The person, shadow, and ray of light form a right-angled triangle just like the triangle formed by the building, shadow, and ray of light; the angles formed must be the same. So we can write a similarity of triangles:
We can isolate the height of the building and calculate it based on known data:
2. A lamp is used to illuminate a room 3 m high between the floor and the ceiling. At a height of 1 m (2 ft) from the floor is a square table with each side measuring 40 cm. Assuming the lamp is a point source located exactly in the center of the table, what is the area of the table's shadow?
In this situation we can analyze the distance between the center of the table and one end. We get the difference between the table and the ceiling equal to 2 m and the average table width equal to 20 cm. Thus we will find the value of x and with it the dimensions of the shadow.
Using similarity of triangles:
We know this is half the shadow dimension, so the total projected dimension is 0.6 m, from where we can calculate the shadow area:
1. An object 20 cm in size is placed at a distance of 4 m from a camera with a hole whose dimension between the entrance and the screen is 50 cm. How big is the projected object on the bulkhead? Will he be inverted?
First we must interpret the problem data. The distance between the object and the camera input is P, the distance between the entrance and the screen is P' and the object size is O. Like this, just apply the darkroom formula:
Isolating the image size, i:
Just apply the values, remembering to use the same unit for all quantities!