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Question

In a model, the distance from the sun to Earth is 7 inches and the distance from the sun to Neptune is 19 inches. If the actual distance from the sun to Earth is 91.549 million miles and the actual distance from the sun to Neptune is 2,780.5 million miles, are the model’s distances proportional to the actual dimensions of the solar system?(1 point)
Responses

Yes, because the ratios can be simplified, and those ratios are equal.
Yes, because the ratios can be simplified, and those ratios are equal.

No, because the model is in inches and the actual distances are in millions of miles.
No, because the model is in inches and the actual distances are in millions of miles.

No, because the ratios of model length to actual distance are different for the two planets.
No, because the ratios of model length to actual distance are different for the two planets.

Yes, because the ratios of model length to actual distance are equal for the two planets.

Answer 1

To determine whether the model's distances are proportional to the actual dimensions of the solar system, we need to calculate the ratios of the model distances to the actual distances for both Earth and Neptune and see if those ratios are equal.

1. Distance from the Sun to Earth:

   • Model distance = 7 inches
   • Actual distance = 91.549 million miles

   Ratio for Earth: \[ \text{Ratio (Earth)} = \frac{\text{Model Distance}}{\text{Actual Distance}} = \frac{7\text{ inches}}{91.549\text{ million miles}} \]

2. Distance from the Sun to Neptune:

   • Model distance = 19 inches
   • Actual distance = 2,780.5 million miles

   Ratio for Neptune: \[ \text{Ratio (Neptune)} = \frac{\text{Model Distance}}{\text{Actual Distance}} = \frac{19\text{ inches}}{2,780.5\text{ million miles}} \]

Now let's calculate these ratios:

1. Calculating the values:
   • Ratio (Earth): \[ \frac{7}{91.549} \approx 0.0764 \text{ inches per million miles} \]
   • Ratio (Neptune): \[ \frac{19}{2,780.5} \approx 0.00684 \text{ inches per million miles} \]

Since \(0.0764 \neq 0.00684\), the ratios of model length to actual distance are not equal for the two planets.

Therefore, the answer is: No, because the ratios of model length to actual distance are different for the two planets.