Question

name:
scale used: 1 to 3.99×10¹⁰

planet	diameter (km)	distance (km)	scaled diameter (cm)	scale drawing of planet	scaled distance from sun (cm)	number of pennies	location along hallway
venus	1.21e4	1.08e8
earth	1.28e4	1.50e8
mars	6.79e3	2.28e8
jupiter	1.43e5	7.78e8
saturn	1.21e5	1.43e9
uranus	5.11e4	2.87e9
neptune	4.95e4	4.50e9
pluto (dwarf planet)	2.3e3	5.90e9

Answer:

To solve for the scaled diameter (in cm) and other related values, we use the scale factor. The scale is \(1\) to \(3.99\times10^{10}\), meaning \(1\) cm on the scale represents \(3.99\times10^{10}\) km in reality. So, to find the scaled diameter (\(d_{scaled}\)) in cm, we use the formula:

\[
d_{scaled} = \frac{\text{Actual Diameter (km)}}{3.99\times10^{10}}
\]

Step 1: Mercury

• Actual Diameter (\(d_{actual}\)): \(4.88\times10^{3}\) km (assuming \(4.88E3\) means \(4.88\times10^{3}\))
• Scaled Diameter:

\[
d_{scaled} = \frac{4.88\times10^{3}}{3.99\times10^{10}} \approx 1.22\times10^{-7} \text{ cm}
\]

Step 2: Venus

• Actual Diameter: \(1.21\times10^{4}\) km (\(1.21E4\))
• Scaled Diameter:

\[
d_{scaled} = \frac{1.21\times10^{4}}{3.99\times10^{10}} \approx 3.03\times10^{-7} \text{ cm}
\]

Step 3: Earth

• Actual Diameter: \(1.28\times10^{4}\) km (\(1.28E4\))
• Scaled Diameter:

\[
d_{scaled} = \frac{1.28\times10^{4}}{3.99\times10^{10}} \approx 3.21\times10^{-7} \text{ cm}
\]

Step 4: Mars

• Actual Diameter: \(6.79\times10^{3}\) km (\(6.79E3\))
• Scaled Diameter:

\[
d_{scaled} = \frac{6.79\times10^{3}}{3.99\times10^{10}} \approx 1.70\times10^{-7} \text{ cm}
\]

Step 5: Jupiter

• Actual Diameter: \(1.43\times10^{5}\) km (\(1.43E5\))
• Scaled Diameter:

\[
d_{scaled} = \frac{1.43\times10^{5}}{3.99\times10^{10}} \approx 3.58\times10^{-6} \text{ cm}
\]

Step 6: Saturn

• Actual Diameter: \(1.21\times10^{5}\) km (\(1.21E5\))
• Scaled Diameter:

\[
d_{scaled} = \frac{1.21\times10^{5}}{3.99\times10^{10}} \approx 3.03\times10^{-6} \text{ cm}
\]

Step 7: Uranus

• Actual Diameter: \(5.11\times10^{4}\) km (\(5.11E4\))
• Scaled Diameter:

\[
d_{scaled} = \frac{5.11\times10^{4}}{3.99\times10^{10}} \approx 1.28\times10^{-6} \text{ cm}
\]

Step 8: Neptune

• Actual Diameter: \(4.95\times10^{4}\) km (\(4.95E4\))
• Scaled Diameter:

\[
d_{scaled} = \frac{4.95\times10^{4}}{3.99\times10^{10}} \approx 1.24\times10^{-6} \text{ cm}
\]

Step 9: Pluto (Dwarf Planet)

• Actual Diameter: \(2.3\times10^{3}\) km (\(2.3E3\))
• Scaled Diameter:

\[
d_{scaled} = \frac{2.3\times10^{3}}{3.99\times10^{10}} \approx 5.76\times10^{-8} \text{ cm}
\]

For the "Scale Distance of Planet" and "Number of Pieces" and "Location along Hallway", we would need more information (like the actual distance from the Sun for each planet) to calculate those. Assuming we have the actual distance from the Sun (\(D_{actual}\)) in km, the scaled distance (\(D_{scaled}\)) in cm would be:

\[
D_{scaled} = \frac{D_{actual}}{3.99\times10^{10}}
\]

And the number of pieces (if, say, each piece is 1 cm) would be approximately equal to the scaled distance in cm. The location along the hallway would be the cumulative scaled distance from the Sun (starting from 0 at the Sun).

Example for Mercury (assuming actual distance from Sun is \(5.79\times10^{7}\) km (\(5.79E7\))):

• Scaled Distance:

\[
D_{scaled} = \frac{5.79\times10^{7}}{3.99\times10^{10}} \approx 1.45\times10^{-3} \text{ cm}
\]

• Number of Pieces: ~\(1.45\times10^{-3}\) (if each piece is 1 cm, this is very small, maybe the scale is different or I misinterpreted the scale. Maybe the scale is \(1\) cm to \(3.99\times10^{10}\) cm? Let's check: \(3.99\times10^{10}\) cm = \(3.99\times10^{8}\) m = \(3.99\times10^{5}\) km. That makes more sense. Let's recalculate with scale \(1\) cm : \(3.99\times10^{5}\) km (since \(3.99\times10^{10}\) cm = \(3.99\times10^{8}\) m = \(3.99\times10^{5}\) km).

Recalculating with correct scale (1 cm : \(3.99\times10^{5}\) km):

Step 1: Mercury (Diameter)

• Actual Diameter: \(4.88\times10^{3}\) km
• Scaled Diameter:

\[
d_{scaled} = \frac{4.88\times10^{3}}{3.99\times10^{5}} \approx 0.0122 \text{ cm}
\]

Step 2: Venus (Diameter)

• Actual Diameter: \(1.21\times10^{4}\) km
• Scaled Diameter:

\[
d_{scaled} = \frac{1.21\times10^{4}}{3.99\times10^{5}} \approx 0.0303 \text{ cm}
\]

Step 3: Earth (Diameter)

• Actual Diameter: \(1.28\times10^{4}\) km
• Scaled Diameter:

\[
d_{scaled} = \frac{1.28\times10^{4}}{3.99\times10^{5}} \approx 0.0321 \text{ cm}
\]

Step 4: Mars (Diameter)

• Actual Diameter: \(6.79\times10^{3}\) km
• Scaled Diameter:

\[
d_{scaled} = \frac{6.79\times10^{3}}{3.99\times10^{5}} \approx 0.0170 \text{ cm}
\]

Step 5: Jupiter (Diameter)

• Actual Diameter: \(1.43\times10^{5}\) km
• Scaled Diameter:

\[
d_{scaled} = \frac{1.43\times10^{5}}{3.99\times10^{5}} \approx 0.358 \text{ cm}
\]

Step 6: Saturn (Diameter)

• Actual Diameter: \(1.21\times10^{5}\) km
• Scaled Diameter:

\[
d_{scaled} = \frac{1.21\times10^{5}}{3.99\times10^{5}} \approx 0.303 \text{ cm}
\]

Step 7: Uranus (Diameter)

• Actual Diameter: \(5.11\times10^{4}\) km
• Scaled Diameter:

\[
d_{scaled} = \frac{5.11\times10^{4}}{3.99\times10^{5}} \approx 0.128 \text{ cm}
\]

Step 8: Neptune (Diameter)

• Actual Diameter: \(4.95\times10^{4}\) km
• Scaled Diameter:

\[
d_{scaled} = \frac{4.95\times10^{4}}{3.99\times10^{5}} \approx 0.124 \text{ cm}
\]

Step 9: Pluto (Dwarf Planet) (Diameter)

• Actual Diameter: \(2.3\times10^{3}\) km
• Scaled Diameter:

\[
d_{scaled} = \frac{2.3\times10^{3}}{3.99\times10^{5}} \approx 0.00576 \text{ cm}
\]

Example for Mercury (Distance from Sun, actual distance \(5.79\times10^{7}\) km):

• Scaled Distance:

\[
D_{scaled} = \frac{5.79\times10^{7}}{3.99\times10^{5}} \approx 145 \text{ cm}
\]

• Number of Pieces (if each piece is 1 cm): ~145
• Location along Hallway: ~145 cm from the Sun.

Example for Venus (Distance from Sun, actual distance \(1.08\times10^{8}\) km):

• Scaled Distance:

\[
D_{scaled} = \frac{1.08\times10^{8}}{3.99\times10^{5}} \approx 271 \text{ cm}
\]

• Number of Pieces: ~271
• Location along Hallway: ~271 cm from the Sun.

Example for Earth (Distance from Sun, actual distance \(1.50\times10^{8}\) km):

• Scaled Distance:

\[
D_{scaled} = \frac{1.50\times10^{8}}{3.99\times10^{5}} \approx 376 \text{ cm}
\]

• Number of Pieces: ~376
• Location along Hallway: ~376 cm from the Sun.

Example for Mars (Distance from Sun, actual distance \(2.28\times10^{8}\) km):

• Scaled Distance:

\[
D_{scaled} = \frac{2.28\times10^{8}}{3.99\times10^{5}} \approx 571 \text{ cm}
\]

• Number of Pieces: ~571
• Location along Hallway: ~571 cm from the Sun.

Example for Jupiter (Distance from Sun, actual distance \(7.78\times10^{8}\) km):

• Scaled Distance:

\[
D_{scaled} = \frac{7.78\times10^{8}}{3.99\times10^{5}} \approx 1949 \text{ cm}
\]

• Number of Pieces: ~1949
• Location along Hallway: ~1949 cm from the Sun.

Example for Saturn (Distance from Sun, actual distance \(1.43\times10^{9}\) km):

• Scaled Distance:

\[
D_{scaled} = \frac{1.43\times10^{9}}{3.99\times10^{5}} \approx 3584 \text{ cm}
\]

• Number of Pieces: ~3584
• Location along Hallway: ~3584 cm from the Sun.

Example for Uranus (Distance from Sun, actual distance \(2.87\times10^{9}\) km):

• Scaled Distance:

\[
D_{scaled} = \frac{2.87\times10^{9}}{3.99\times10^{5}} \approx 7193 \text{ cm}
\]

• Number of Pieces: ~7193
• Location along Hallway: ~7193 cm from the Sun.

Example for Neptune (Distance from Sun, actual distance \(4.50\times10^{9}\) km):

• Scaled Distance:

\[
D_{scaled} = \frac{4.50\times10^{9}}{3.99\times10^{5}} \approx 11278 \text{ cm}
\]

• Number of Pieces: ~11278
• Location along Hallway: ~11278 cm from the Sun.

Example for Pluto (Distance from Sun, actual distance \(5.90\times10^{9}\) km):

• Scaled Distance:

\[
D_{scaled} = \frac{5.90\times10^{9}}{3.99\times10^{5}} \approx 14787 \text{ cm}
\]

• Number of Pieces: ~14787
• Location along Hallway: ~14787 cm from the Sun.

These calculations assume the scale is \(1\) cm to \(3.99\times10^{5}\) km (correcting the initial misinterpretation of the scale unit). If the scale is indeed \(1\) to \(3.99\times10^{10}\) (with the same unit, e.g., km), the values would be much smaller as calculated initially. The key is to use the correct scale factor (actual value / scale factor) to find the scaled value.