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
To determine if the model's distances are proportional to the actual dimensions of the solar system, we can set up a proportion.
Let's define the scale factor (SF) as the ratio of the model distances to the actual distances.
For the distance from the sun to Earth:
SF = (Model distance from sun to Earth) / (Actual distance from sun to Earth)
SF = 7 inches / 91.549 million miles
For the distance from the sun to Neptune:
SF = (Model distance from sun to Neptune) / (Actual distance from sun to Neptune)
SF = 19 inches / 2780.5 million miles
To determine if the model's distances are proportional to the actual dimensions, we need to compare the scale factors. If the scale factors are equal, then the distances are proportional.
Comparing the two scale factors:
SF1/SF2 = (7 inches / 91.549 million miles) / (19 inches / 2780.5 million miles)
SF1/SF2 = (7 inches * 2780.5 million miles) / (19 inches * 91.549 million miles)
Simplifying the expression:
SF1/SF2 = (19553.5 inches * million miles) / (1721.831 million miles * inches)
The units cancel out:
SF1/SF2 = 19553.5 / 1721.831
Calculating the value:
SF1/SF2 ≈ 11.36
Since the scale factor ratio is not equal to 1, the model's distances are not proportional to the actual dimensions of the solar system.
To determine if the model's distances are proportional to the actual dimensions of the solar system, we need to compare the ratios of the model's distances to the actual distances for both Earth and Neptune.
Let's calculate the ratio for Earth:
Model distance from Sun to Earth: 7 inches
Actual distance from Sun to Earth: 91.549 million miles
Ratio = Model distance / Actual distance
Ratio = 7 inches / 91.549 million miles
We can simplify the units by converting inches to miles. Since there are 63,360 inches in a mile, we divide the model distance by this conversion factor:
Ratio = (7 inches / 63,360 inches) / 91.549 million miles
Ratio = (7 / 63,360) / 91.549 million
Ratio ≈ 9.13 x 10^-11
Now let's calculate the ratio for Neptune:
Model distance from Sun to Neptune: 19 inches
Actual distance from Sun to Neptune: 2,780.5 million miles
Ratio = Model distance / Actual distance
Ratio = 19 inches / 2,780.5 million miles
Again, we convert inches to miles:
Ratio = (19 inches / 63,360 inches) / 2,780.5 million miles
Ratio = (19 / 63,360) / 2,780.5 million
Ratio ≈ 1.11 x 10^-10
Comparing the two ratios, we can see that they are not equal. The ratio for Neptune is slightly larger than the ratio for Earth.
Therefore, we can conclude that the model's distances are not perfectly proportional to the actual dimensions of the solar system.
To determine if the model's distances are proportional to the actual dimensions of the solar system, we can compare the ratios of the model distances to the actual distances.
The ratio of the model distance from the sun to Earth (7 inches) to the actual distance from the sun to Earth (91.549 million miles) is:
(7 inches) / (91.549 million miles)
The ratio of the model distance from the sun to Neptune (19 inches) to the actual distance from the sun to Neptune (2,780.5 million miles) is:
(19 inches) / (2,780.5 million miles)
To determine if these ratios are equal, we can simplify them by converting the units of measurement for the model distances and comparing them to the actual distances.
Let's convert the model distances from inches to miles:
- 7 inches is equivalent to (7 inches / 12 inches/foot) * (1 foot/5280 feet/mile) = (7/12) * (1/5280) miles = 0.001325757 miles.
- 19 inches is equivalent to (19 inches / 12 inches/foot) * (1 foot/5280 feet/mile) = (19/12) * (1/5280) miles = 0.003586742 miles.
Now we can compare the simplified ratios:
- (0.001325757 miles) / (91.549 million miles)
- (0.003586742 miles) / (2,780.5 million miles)
By calculating these ratios, we can determine if they are roughly equal or differ significantly.