### Brain Teasers

# Solar System Model

Kate and Shireen are making a model of the solar system on the ceiling of their classroom. They plan to place their model of the Sun at the southwest corner of the ceiling, and they will mark out the relative distances of the planets along a line from the southwest corner to the northeast corner. The classroom measures 42 ft by 66 ft.

In order to make the best use of the ceiling space, the girls decide to place their model of Pluto at the northeast corner. They understand that placing Pluto there automatically determines the scale of their model. But in order to know where to place the other planets, the girls need to know what that scale is.

Assuming Pluto, the farthest planet, is 5.9 * 10^9 km from the Sun, approximately what distance in the real solar system does 1 centimeter on the model now represent?

In order to make the best use of the ceiling space, the girls decide to place their model of Pluto at the northeast corner. They understand that placing Pluto there automatically determines the scale of their model. But in order to know where to place the other planets, the girls need to know what that scale is.

Assuming Pluto, the farthest planet, is 5.9 * 10^9 km from the Sun, approximately what distance in the real solar system does 1 centimeter on the model now represent?

### Hint

1. Use the Pythagorean Theorem.2. To convert feet to centimeters, multiply by 30.48.

### Answer

On the solar system model, 1 cm represents approximately 2.5 * 10^6 km in the real solar system.Since the room measures 42 ft by 66 ft, the diagonal distance between the southwest and northeast corners, by the Pythagorean Theorem, is the square root of (42^2 + 66^2) sqft = 78 ft (rounded) or 2377 cm.

So, if 2377 cm in the model represents a real distance of 5900 * 10^6 km, then 1 cm represents 5900/2377 * 10^6 km or 2.5 * 10^6 km.

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