Question

part a

the table shows the diameters of the planets in our solar system. assume that a basketball whose diameter is 25 centimeters represents the planet jupiter. find the scale (ratio) between jupiter and the basketball. then use this ratio to find the scaled diameter of the other planets. enter these numbers into the table. (saturn has been done for you.) finally, choose a real-world spherical or nearly spherical object that matches the scaled diameter of each planet.

saturn is shown as an example. to find what saturns size would be after adjusting it to scale, follow these steps. let the scaled diameter of saturn be \\(d\\) centimeters. if jupiters diameter, 142,984 kilometers, is scaled to 25 centimeters, saturns diameter of 120,536 kilometers will be scaled to about 21 centimeters:

\\\frac{142,984}{25} = \frac{120,536}{d}\\
\\d = \frac{120,536 \times 25}{142,984}\\
\\d \approx 21\text{ cm}\\

use this process to complete the table.

	mercury	venus	earth	mars	jupiter	saturn	uranus	neptune
scaled diameter (cm)					25	21
spherical object					basketball	deflated basketball

Explanation:

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<reasoning>

Calculate the scaling ratio

We are given that Jupiter's actual diameter is \(142,984\text{ km}\) and its scaled diameter is \(25\text{ cm}\).
The scaling ratio \(k\) is:
\[k = \frac{25}{142,984}\text{ cm/km}\]

Calculate scaled diameters

Using Scientific Modeling, we scale each planet's diameter \(D\) using the formula \(d = D \times \frac{25}{142,984}\):

• Mercury: \(4,879 \times \frac{25}{142,984} \approx 0.85\text{ cm}\) (rounds to \(0.9\text{ cm}\) or \(0.85\text{ cm}\))
• Venus: \(12,104 \times \frac{25}{142,984} \approx 2.12\text{ cm}\) (rounds to \(2.1\text{ cm}\))
• Earth: \(12,756 \times \frac{25}{142,984} \approx 2.23\text{ cm}\) (rounds to \(2.2\text{ cm}\))
• Mars: \(6,792 \times \frac{25}{142,984} \approx 1.19\text{ cm}\) (rounds to \(1.2\text{ cm}\))
• Uranus: \(51,118 \times \frac{25}{142,984} \approx 8.94\text{ cm}\) (rounds to \(8.9\text{ cm}\))
• Neptune: \(49,528 \times \frac{25}{142,984} \approx 8.66\text{ cm}\) (rounds to \(8.7\text{ cm}\))

Select matching spherical objects

We choose real-world spherical objects that match these scaled diameters:

• Mercury (\(\approx 0.9\text{ cm}\)): Peppercorn or small blueberry
• Venus (\(\approx 2.1\text{ cm}\)): Nickel or grape
• Earth (\(\approx 2.2\text{ cm}\)): Nickel or cherry
• Mars (\(\approx 1.2\text{ cm}\)): Blueberry or marble
• Uranus (\(\approx 8.9\text{ cm}\)): Baseball or tennis ball
• Neptune (\(\approx 8.7\text{ cm}\)): Baseball or tennis ball

</reasoning>

<answer>

Planet	Mercury	Venus	Earth	Mars	Jupiter	Saturn	Uranus	Neptune
Scaled diameter (cm)	0.9	2.1	2.2	1.2	25	21	8.9	8.7
Spherical object	peppercorn	grape	nickel	blueberry	basketball	deflated basketball	baseball	tennis ball

</answer>

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Answer:

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<reasoning>

Calculate the scaling ratio

We are given that Jupiter's actual diameter is \(142,984\text{ km}\) and its scaled diameter is \(25\text{ cm}\).
The scaling ratio \(k\) is:
\[k = \frac{25}{142,984}\text{ cm/km}\]

Calculate scaled diameters

Using Scientific Modeling, we scale each planet's diameter \(D\) using the formula \(d = D \times \frac{25}{142,984}\):

• Mercury: \(4,879 \times \frac{25}{142,984} \approx 0.85\text{ cm}\) (rounds to \(0.9\text{ cm}\) or \(0.85\text{ cm}\))
• Venus: \(12,104 \times \frac{25}{142,984} \approx 2.12\text{ cm}\) (rounds to \(2.1\text{ cm}\))
• Earth: \(12,756 \times \frac{25}{142,984} \approx 2.23\text{ cm}\) (rounds to \(2.2\text{ cm}\))
• Mars: \(6,792 \times \frac{25}{142,984} \approx 1.19\text{ cm}\) (rounds to \(1.2\text{ cm}\))
• Uranus: \(51,118 \times \frac{25}{142,984} \approx 8.94\text{ cm}\) (rounds to \(8.9\text{ cm}\))
• Neptune: \(49,528 \times \frac{25}{142,984} \approx 8.66\text{ cm}\) (rounds to \(8.7\text{ cm}\))

Select matching spherical objects

We choose real-world spherical objects that match these scaled diameters:

• Mercury (\(\approx 0.9\text{ cm}\)): Peppercorn or small blueberry
• Venus (\(\approx 2.1\text{ cm}\)): Nickel or grape
• Earth (\(\approx 2.2\text{ cm}\)): Nickel or cherry
• Mars (\(\approx 1.2\text{ cm}\)): Blueberry or marble
• Uranus (\(\approx 8.9\text{ cm}\)): Baseball or tennis ball
• Neptune (\(\approx 8.7\text{ cm}\)): Baseball or tennis ball

</reasoning>

<answer>

Planet	Mercury	Venus	Earth	Mars	Jupiter	Saturn	Uranus	Neptune
Scaled diameter (cm)	0.9	2.1	2.2	1.2	25	21	8.9	8.7
Spherical object	peppercorn	grape	nickel	blueberry	basketball	deflated basketball	baseball	tennis ball

</answer>

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