When you look into your car's 5.0-cm-tall rear-view mirror from 35 cm away, the front of a bus, from the ground to the roof, exactly fills the mirror. If the bus is 17 m from the mirror, how tall is the bus?

검증된 단계별 안내

Understand the problem: The rear-view mirror acts as a plane mirror, and the height of the image in the mirror is proportional to the height of the object (the bus) based on the geometry of similar triangles. We are tasked with finding the height of the bus.

Set up the relationship using similar triangles: The height of the image in the mirror (5.0 cm) and the height of the bus (h) are proportional to their respective distances from the mirror. The distance of the image is the distance from the observer to the mirror (35 cm), and the distance of the bus is 17 m (convert to cm: 1700 cm).

Write the proportionality equation: \( \frac{\text{Height of image}}{\text{Distance to image}} = \frac{\text{Height of object}}{\text{Distance to object}} \). Substituting the known values, \( \frac{5.0}{35} = \frac{h}{1700} \).

Solve for the height of the bus (h): Rearrange the equation to isolate \( h \): \( h = \frac{5.0 \times 1700}{35} \).

Perform the calculation to find \( h \): Simplify the expression \( h = \frac{5.0 \times 1700}{35} \) to determine the height of the bus in cm, and convert the result to meters if needed.

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주요 개념

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

Similar Triangles

The concept of similar triangles states that if two triangles have the same shape, their corresponding angles are equal, and the lengths of their corresponding sides are proportional. In this scenario, the rear-view mirror and the bus form two similar triangles, allowing us to set up a proportion to find the height of the bus based on the height of the mirror and the distances involved.

Proportional Relationships

Proportional relationships occur when two quantities maintain a constant ratio. In this problem, the height of the bus and the height of the mirror can be related through their respective distances from the observer. By establishing a ratio between the height of the mirror and the height of the bus, we can solve for the unknown height of the bus using the distances provided.

Distance and Magnification

Distance and magnification are key in understanding how objects appear in mirrors. The apparent size of an object in a mirror is influenced by the distance from the observer to the mirror and the object itself. In this case, the distances from the mirror to the observer and the bus help determine how the bus's height can be calculated based on the mirror's height and the distances involved.