You wish to project the image of a slide on a screen 9.00 m from the lens of a slide projector. If the dimensions of the picture on a 35 mm color slide are 24 mm ✖ 36 mm, what is the minimum size of the projector screen required to accommodate the image?

검증된 단계별 안내

Determine the magnification (M) of the image using the lens formula and the given distances. The magnification is given by \( M = \frac{d_i}{d_o} \), where \( d_i \) is the image distance (9.00 m or 9000 mm) and \( d_o \) is the object distance (distance from the slide to the lens).

Calculate the dimensions of the projected image on the screen using the magnification. The width of the image is \( M \times 36 \ \text{mm} \), and the height of the image is \( M \times 24 \ \text{mm} \).

Convert the dimensions of the projected image from millimeters to meters by dividing by 1000, as the screen size is typically measured in meters.

Determine the minimum size of the projector screen required by considering the larger of the two dimensions (width or height) and ensuring the screen can accommodate both dimensions.

Summarize the result by stating the minimum dimensions of the screen in meters, ensuring it is large enough to display the entire projected image.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Magnification

Magnification is the process of enlarging the appearance of an object through optical devices like lenses. It is defined as the ratio of the image size to the object size. In this context, understanding how the slide's dimensions relate to the projected image size on the screen is crucial for determining the minimum screen size needed.

Lens Formula

The lens formula relates the object distance (u), image distance (v), and focal length (f) of a lens, expressed as 1/f = 1/v + 1/u. This formula is essential for calculating how the slide's image is formed by the projector lens, which directly influences the size of the projected image on the screen.

Aspect Ratio

Aspect ratio is the ratio of the width to the height of an image or screen. For the slide dimensions given (24 mm by 36 mm), calculating the aspect ratio helps ensure that the projected image maintains its proportions on the screen. This is important for determining the minimum dimensions of the screen to avoid distortion of the image.