To solve the problem of determining where an object should be placed in front of a pinhole camera such that the image is diminished to half of its height, we follow these steps:
1. Understand the Pin Hole Camera Model:
The pinhole camera model is based on similar triangles. In this model, the ratio of the object's distance to its height is equal to the ratio of the image's distance to its height.
2. Given Information:
- The width of the pinhole camera is [tex]\(0.2 \, \text{meters}\)[/tex]. This width can be interpreted as the distance from the pinhole (the aperture) to the image plane, known as the image distance.
- The image is diminished to half its height, so the image height is half the object height. Mathematically, if the object's height is [tex]\(H\)[/tex], then the image height is [tex]\( \frac{H}{2} \)[/tex].
3. Set Up the Proportions:
Using the pinhole camera model, the relationship between the object distance ([tex]\(D\)[/tex]), the object height ([tex]\(H\)[/tex]), the image distance ([tex]\(d_i\)[/tex]), and the image height ([tex]\(h_i\)[/tex]) can be described by the following proportion:
[tex]\[
\frac{D}{H} = \frac{d_i}{h_i}
\][/tex]
Given that [tex]\(d_i = 0.2 \, \text{meters}\)[/tex] and [tex]\(h_i = \frac{H}{2}\)[/tex], we can substitute these values into the proportion:
[tex]\[
\frac{D}{H} = \frac{0.2 \, \text{meters}}{\frac{H}{2}}
\][/tex]
4. Solve for Object Distance [tex]\(D\)[/tex]:
Simplify the proportion:
[tex]\[
\frac{D}{H} = \frac{0.2 \, \text{meters}}{\frac{H}{2}} = \frac{0.2 \, \text{meters}}{0.5H} = \frac{0.2 \, \text{meters}}{\frac{1}{2}H} = 0.4
\][/tex]
So, the object distance [tex]\(D\)[/tex] can be calculated as:
[tex]\[
D = 0.4 \, \text{meters}
\][/tex]
Answer:
The object should be placed [tex]\(0.4 \, \text{meters}\)[/tex] in front of the pinhole camera to produce an image that is diminished to half of its height.
