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 91,549,000 miles in the actual distance from the sun, Neptune is 2780,500,000 miles are the models distance proportional to the actual dimensions of the solar system?
a. Yes because the ratios of model length to actual distance are equal to the two planets
b. Yes because the ratios can be simplified, and those ratios are equal
C. no because the ratios of model length to actual distance are different from the two planets
D. no because the model is in inches in the actual distance are in millions of miles

Answer 1

To determine if the model distances are proportional to the actual distances, we can examine the ratios of the model distances to the actual distances for both Earth and Neptune.

1. **Calculate the ratio for Earth:**
- Model distance from the Sun to Earth = 7 inches
- Actual distance from the Sun to Earth = 91,549,000 miles

Ratio for Earth:
\[
\text{Ratio}_{\text{Earth}} = \frac{\text{Model Distance}}{\text{Actual Distance}} = \frac{7 \text{ inches}}{91,549,000 \text{ miles}}
\]

2. **Calculate the ratio for Neptune:**
- Model distance from the Sun to Neptune = 19 inches
- Actual distance from the Sun to Neptune = 2,780,500,000 miles

Ratio for Neptune:
\[
\text{Ratio}_{\text{Neptune}} = \frac{19 \text{ inches}}{2,780,500,000 \text{ miles}}
\]

3. **Compare the two ratios:**
- To compare the ratios, we can put them in a similar form (often done by finding a common unit, which in this case we can compare their ratios directly, even if they are in different units):
\[
\text{Ratio}_{\text{Earth}} \approx \frac{7}{91,549,000}
\]
\[
\text{Ratio}_{\text{Neptune}} \approx \frac{19}{2,780,500,000}
\]

Now let's compute the values:
- For Earth:
\[
\frac{7}{91,549,000} \approx 7.63 \times 10^{-8}
\]
- For Neptune:
\[
\frac{19}{2,780,500,000} \approx 6.84 \times 10^{-9}
\]

Since \( 7.63 \times 10^{-8} \) is not equal to \( 6.84 \times 10^{-9} \), the two ratios are not equal.

Thus, we can conclude that the model distances are not proportional to the actual dimensions of the solar system.

The correct answer is:
**C. no because the ratios of model length to actual distance are different from the two planets.**