Question

Scale of the Solar System. The real diameters of the Sun and Earth are approximately 1.4 million kilometers and 12,800 kilometers, respectively. The Earth-Sun distance is approximately 150 million kilometers. Calculate the sizes of Earth and the Sun. and the distance between them, on a scale of 1 to 10 billion. Show your work clearly.

Short answer

Sun: 0.00014 km, Earth: 0.00000128 km, Distance: 0.015 km on the scale.

Step-by-step solution

Understand the Scale

We are given a scale of 1 to 10 billion, meaning that in our model, 1 unit of distance will represent 10 billion units of real distance. Therefore, we need to divide each real measurement by 10 billion to get the modeled sizes and distances.

Calculate the Scaled Diameter of the Sun

The real diameter of the Sun is approximately 1.4 million kilometers. To find the scaled diameter, divide by 10 billion: \(\frac{1,400,000}{10,000,000,000} = 0.00014\) kilometers.

Calculate the Scaled Diameter of Earth

The real diameter of Earth is approximately 12,800 kilometers. To find the scaled diameter, divide by 10 billion: \(\frac{12,800}{10,000,000,000} = 0.00000128\) kilometers.

Calculate the Scaled Distance Between Earth and the Sun

The real distance from Earth to the Sun is approximately 150 million kilometers. To find the scaled distance, divide by 10 billion: \(\frac{150,000,000}{10,000,000,000} = 0.015\) kilometers.

Key concepts

Mathematical Scaling

Mathematical scaling is like a magic tool that makes enormous distances and sizes more manageable. Think of it as shrinking the universe down to fit into our models. This concept is crucial when working with astronomical objects, which are beyond vast. Everything in real life is huge, so we use scaling to create models that help us understand and explore them.
For example, in the solar system scale model, we use a scale of 1 to 10 billion. This means every real-world kilometer becomes a minuscule fraction in our model, making it possible to visualize and compare distances and sizes that are otherwise too large to grasp.
By dividing the real measurements by 10 billion, we simplify immense values into something more relatable. So, when we calculate the scaled diameter of the Sun or the Earth, we're just turning these massive numbers into tiny, understandable figures that we can handle. It's like putting the entire universe on a small canvas without losing its beauty.

Solar System Distances

Distances in our solar system are mind-boggling. The Earth and the Sun itself are separated by 150 million kilometers! Such numbers can be tough to wrap our heads around.
The Sun's diameter is about 1.4 million kilometers, while Earth's diameter is around 12,800 kilometers. But, how do these immense distances translate into a manageable form? That's where scaling comes into play.
For our scale model at 1 to 10 billion, the Earth-Sun distance shrinks from an unfathomable expanse to just 0.015 kilometers, which feels much more tangible. Similarly, the Sun and Earth's diameters become very small figures.

• The scaled Sun diameter: 0.00014 kilometers.
• The scaled Earth diameter: 0.00000128 kilometers.

Scaling these measurements allows us to compare distances like never before, making the solar system's vastness a little easier to comprehend.

Astronomical Measurements

Astronomical measurements let us capture the grand scale of celestial bodies and distances with precision. They're essential for understanding our universe's layout and the entities within it.
We employ models and measurements to get a sense of distances, sizes, and the relationships between celestial bodies. This real-world data is vital because it forms the backbone of our scale models.
When measuring astronomical distances, units like kilometers become cumbersome due to their enormity. Instead, using scaling, we convert these measurements so that they're easier to work with.
To express vast space objects and distances effectively, astronomers have devised different ways of measurement and scaling is just one of the ingenious methods employed. As a significant part of this discipline, dividing real-world values by a constant like 10 billion, as in our example, makes them manageable and facilitates learning and visualization. This way, even the biggest entities can be understood on a human level.