In a model the dish 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 models distances proportional to the actual dimensions of the solar system
To determine if the model distances are proportional to the actual dimensions of the solar system, we need to compare the ratios of the model distances to the actual distances for both Earth and Neptune.
The ratio for Earth would be:
Model distance for Earth / Actual distance for Earth = 7 inches / 91.549 million miles.
The ratio for Neptune would be:
Model distance for Neptune / Actual distance for Neptune = 19 inches / 2,780.5 million miles.
Now we can simplify these ratios:
Model distance for Earth / Actual distance for Earth = 7 inches / 91.549 million miles = 7/91.549 * 1/1 million = 0.0000764 inches per mile.
Model distance for Neptune / Actual distance for Neptune = 19 inches / 2,780.5 million miles = 19/2,780.5 * 1/1 million = 0.00000683 inches per mile.
Comparing these ratios, we can see that the ratio for the model distance to the actual distance is different for Earth and Neptune. Therefore, the model distances are not proportional to the actual dimensions of the solar system.
To determine if the model distances are proportional to the actual dimensions of the solar system, we need to compare the ratios of the model distances to the actual distances for both Earth and Neptune.
Let's calculate the ratios:
For Earth:
- Model distance: 7 inches
- Actual distance: 91.549 million miles
Ratio = Model distance / Actual distance = 7 inches / 91.549 million miles
For Neptune:
- Model distance: 19 inches
- Actual distance: 2,780.5 million miles
Ratio = Model distance / Actual distance = 19 inches / 2,780.5 million miles
To check if the ratios are proportional, we compare the two ratios. If they are equal, then the model distances are proportional to the actual distances.
Ratio for Earth: 7 inches / 91.549 million miles
Ratio for Neptune: 19 inches / 2,780.5 million miles
To compare these ratios, we can express the ratios in the same units:
Ratio for Earth: (7 inches) / (91.549 million miles) = (7 inches) / (91.549 million miles) * (1,000,000) / (1,000,000) = (7 inches * 1,000,000) / (91.549 million miles) = 7,000,000 inches / 91.549 million miles
Ratio for Neptune: (19 inches) / (2,780.5 million miles) = (19 inches) / (2,780.5 million miles) * (1,000,000) / (1,000,000) = (19 inches * 1,000,000) / (2,780.5 million miles) = 19,000,000 inches / 2,780.5 million miles
Now we can compare the two ratios:
Ratio for Earth: 7,000,000 inches / 91.549 million miles
Ratio for Neptune: 19,000,000 inches / 2,780.5 million miles
If these two ratios are approximately equal, then the model distances are proportional to the actual distances in the solar system.
To determine if the model distances are proportional to the actual dimensions of the solar system, we need to compare the ratios of the model distances to the actual distances for both Earth-Sun and Neptune-Sun.
Let's start by calculating the ratios:
Ratio for Earth-Sun:
Model distance: 7 inches
Actual distance: 91.549 million miles
Ratio = Model distance / Actual distance
Ratio = 7 inches / 91.549 million miles
Ratio for Neptune-Sun:
Model distance: 19 inches
Actual distance: 2,780.5 million miles
Ratio = Model distance / Actual distance
Ratio = 19 inches / 2,780.5 million miles
Now, let's simplify both ratios:
Ratio for Earth-Sun:
Ratio ≈ 7 / 91,549
Ratio for Neptune-Sun:
Ratio ≈ 19 / 2,780,500
To check if the model distances are proportional to the actual dimensions, we need to see if these simplified ratios are equal.
Ratio for Earth-Sun ≈ 7 / 91,549
Ratio for Neptune-Sun ≈ 19 / 2,780,500
Since these ratios are not equal, the model distances are not proportional to the actual dimensions of the solar system.