Question

Translate to an equation and solve. The sun is a sphere with a radius of about \(695,990\) kilometers. The core of the sun is a sphere with a radius of about \(170,000\) kilometers. What percent of the sun's total volume is made up of the core?

Short answer

Answer: Approximately 0.287%

Step-by-step solution

Find the volume of the sun

Using the formula for the volume of a sphere, \(V = \frac{4}{3}\pi r^3\), where 'V' is the volume and 'r' is the radius, we can calculate the volume of the sun. Given that the radius of the sun is \(695,990\) kilometers, the volume can be calculated as follows: \(V_{sun} = \frac{4}{3}\pi (695,990)^3\).

Find the volume of the core

Using the same formula as in step 1, we can calculate the volume of the core. The radius of the core is given as \(170,000\) kilometers, so the calculation will be: \(V_{core} = \frac{4}{3}\pi (170,000)^3\).

Divide the volume of the core by the volume of the sun

In order to find the percentage of the core volume in relation to the total sun volume, we need to divide the volume of the core by the volume of the sun and then multiply the result by 100. The formula is as follows: \(Percentage = \frac{V_{core}}{V_{sun}} × 100\).

Substitute the values and calculate the percentage

Plugging in the values we found in steps 1 and 2, we get: \(Percentage = \frac{\frac{4}{3}\pi (170,000)^3}{\frac{4}{3}\pi (695,990)^3} × 100\) Notice that the \(\frac{4}{3}\pi\) term can be canceled out in both numerator and denominator: \(Percentage = \frac{(170,000)^3}{(695,990)^3} × 100\) Now, we can calculate the percentage: \(Percentage \approx 0.287 \%\) So, approximately \(0.287 \%\) of the sun's total volume is made up of its core.

Key concepts

Sphere Volume Calculation

Understanding the computation of a sphere's volume is essential for solving a variety of geometric problems. The volume of a sphere is given by the formula \( V = \frac{4}{3}\pi r^3 \), where \( V \) represents the volume and \( r \) is the radius of the sphere. This formula arises from calculus and represents the integral of concentric shells filling the entire sphere.

To calculate the volume of a sphere, follow these steps:

• Measure the radius (the distance from the center of the sphere to any point on its surface).
• Cube the radius (multiply it by itself twice).
• Multiply the cubed radius by \( \pi \) (approximately 3.14159).
• Multiply this product by \( \frac{4}{3} \).

Thus, the larger the radius, the greater the volume of the sphere.

Percentage Calculation

Calculating percentages is a fundamental math skill that allows you to express a part as a fraction of 100. To determine what percentage one quantity is of another, divide the part by the whole and then multiply by 100. This is often represented as \( \frac{part}{whole} \) followed by \( \times 100 \).

The steps for percentage calculation include:

• Determining the 'part' or 'portion' you wish to compare.
• Determining the 'whole' or the total context of the comparison.
• Dividing the 'part' by the 'whole' to find the fraction.
• Multiplying the fraction by 100 to convert it to a percentage.

For example, to find what percentage the volume of the core is of the sun's total volume, divide the volume of the core by the volume of the sun, then multiply by 100, as shown in the exercise solution.

Radius and Volume Relationship

There is a powerful relationship between the radius of a sphere and its volume. The volume of a sphere varies with the cube of its radius, as shown in the volume formula \( V = \frac{4}{3}\pi r^3 \). This implies that minor changes in radius cause significant changes in volume.

For instance, if the radius is doubled, the volume increases by a factor of eight (since \( 2^3 = 8 \) ). Similarly, decreasing the radius by half results in a reduction of volume to one-eighth of its original size. This is crucial to understanding the solar core's volume in comparison to the sun's volume. Even though the core radius is a significant measurement, it forms a surprisingly small percentage of the sun's total volume due to this cubic relationship.