A student has built a 15-cm-long pinhole camera for a science fair project. She wants to photograph her 180-cm-tall friend and have the image on the film be 5.0 cm high. How far should the front of the camera be from her friend?

Verified step by step guidance

1. Understand the problem: This is a case of similar triangles formed by the object (the friend) and its image (on the film inside the pinhole camera). The height of the object, the height of the image, and the distances from the pinhole to the object and the image are related proportionally.
2. Write the relationship for similar triangles: \( \frac{h_{\text{image}}}{h_{\text{object}}} = \frac{d_{\text{image}}}{d_{\text{object}}} \), where \( h_{\text{image}} = 5.0 \ \text{cm} \), \( h_{\text{object}} = 180 \ \text{cm} \), and \( d_{\text{image}} = 15 \ \text{cm} \) (the length of the pinhole camera).
3. Rearrange the formula to solve for \( d_{\text{object}} \): \( d_{\text{object}} = \frac{d_{\text{image}} \cdot h_{\text{object}}}{h_{\text{image}}} \).
4. Substitute the known values into the equation: \( d_{\text{object}} = \frac{15 \cdot 180}{5.0} \).
5. Simplify the expression to find \( d_{\text{object}} \), which represents the distance from the front of the camera to the friend. This will give the required distance.

Key Concepts

Similar Triangles

The concept of similar triangles is fundamental in optics, particularly in pinhole cameras. When light rays pass through the pinhole, they create an inverted image on the film. The relationship between the height of the object, the height of the image, and their respective distances from the pinhole can be analyzed using the properties of similar triangles, which states that corresponding angles are equal and the ratios of corresponding sides are proportional.

Magnification

Magnification in optics refers to the ratio of the height of the image to the height of the object. It is a crucial concept for understanding how the size of an image changes based on the distance from the object to the camera. In this scenario, the magnification can be calculated using the formula: magnification = height of image / height of object, which helps determine the necessary distance to achieve the desired image size.

Pinhole Camera Formula

The pinhole camera formula relates the distances of the object and image to the size of the image produced. It can be expressed as: (height of image / height of object) = (distance from pinhole to image / distance from pinhole to object). This formula allows the student to calculate how far the camera should be positioned from her friend to achieve the desired image size, integrating the concepts of similar triangles and magnification.