In order to measure the length of a long flourescent ceiling lamp, a student makes a very small hole in a piece of paper, holds the paper above the floor, measures the distance between the hole and the floor, and finally measures the length of the light's image on the floor.
When the distance between the hole and floor is h1=20 cm, the image's length is l1=20 cm; when the distance between the hole and floor is h2=33.333 cm, the image's length is l2=36 cm . What is the length L of the ceiling lamp in meters?
To find the length of the fluorescent ceiling lamp, we can use similar triangles.
First, let's define the variables:
- l1: length of the image on the floor when the distance between the hole and floor is h1.
- h1: distance between the hole and floor when the image length is l1.
- l2: length of the image on the floor when the distance between the hole and floor is h2.
- h2: distance between the hole and floor when the image length is l2.
- L: length of the ceiling lamp (what we're trying to find).
We can set up a proportion using the similar triangles formed by the lamp and its image:
(l1 / h1) = (l2 / h2)
Now, we substitute the given values:
(20 cm / 20 cm) = (36 cm / 33.333 cm)
We can cross-multiply and solve for l1:
20 cm * 33.333 cm = 36 cm * 20 cm
666.66 cm^2 = 720 cm^2
Now, we know that the length of the lamp L is equal to h1 multiplied by the ratio of l1 and h1:
L = (h1 * l1) / h2
Substituting the given values:
L = (20 cm * l1) / 33.333 cm
Since we've already found that l1 = 20 cm, we substitute it:
L = (20 cm * 20 cm) / 33.333 cm
Finally, we convert the length from centimeters to meters by dividing by 100:
L = (20 cm * 20 cm) / (33.333 cm * 100) = 0.120012 cm^2
Therefore, the length L of the fluorescent ceiling lamp is approximately 0.120012 meters.