Question
For each planet in a solar system, its year is the time it takes the planet to revolve around the center star. The formula Upper E equals 0.2 x Superscript 3 divided by 2 models the number of Earth days in a planet's year, E, where x is the average distance of the planet from the center star, in millions of kilometers. There are approximately 224.2 Earth days in the year of Planet Upper B. What is the average distance of Planet Upper B from the center star?
Question content area bottom
Part 1
The average distance is approximately
enter your response here million kilometers.
(Do not round until the final answer. Then round to the nearest whole number as needed.)
Question content area bottom
Part 1
The average distance is approximately
enter your response here million kilometers.
(Do not round until the final answer. Then round to the nearest whole number as needed.)
Answers
GPT-4o mini
To find the average distance of Planet B from the center star, we use the formula given:
\[
E = 0.2 \cdot x^{3/2}
\]
where \( E \) is the number of Earth days in the planet's year, and \( x \) is the average distance in millions of kilometers.
Given that \( E = 224.2 \) days for Planet B, we can substitute this value into the equation:
\[
224.2 = 0.2 \cdot x^{3/2}
\]
To isolate \( x^{3/2} \), we'll first divide both sides by 0.2:
\[
x^{3/2} = \frac{224.2}{0.2}
\]
Calculating the right side:
\[
x^{3/2} = 1121
\]
Next, we need to solve for \( x \). To do this, we will raise both sides of the equation to the power of \( \frac{2}{3} \):
\[
x = (1121)^{\frac{2}{3}}
\]
Now we calculate \( (1121)^{\frac{2}{3}} \). First, find the cube root of 1121 and then square the result.
Calculating the cube root of 1121:
\[
\sqrt[3]{1121} \approx 10.513
\]
Now squaring this result:
\[
x \approx (10.513)^2 \approx 110.56
\]
Finally, rounding to the nearest whole number gives:
\[
x \approx 111
\]
Thus, the average distance of Planet B from the center star is approximately **111 million kilometers**.
**Final Answer: 111 million kilometers**.
\[
E = 0.2 \cdot x^{3/2}
\]
where \( E \) is the number of Earth days in the planet's year, and \( x \) is the average distance in millions of kilometers.
Given that \( E = 224.2 \) days for Planet B, we can substitute this value into the equation:
\[
224.2 = 0.2 \cdot x^{3/2}
\]
To isolate \( x^{3/2} \), we'll first divide both sides by 0.2:
\[
x^{3/2} = \frac{224.2}{0.2}
\]
Calculating the right side:
\[
x^{3/2} = 1121
\]
Next, we need to solve for \( x \). To do this, we will raise both sides of the equation to the power of \( \frac{2}{3} \):
\[
x = (1121)^{\frac{2}{3}}
\]
Now we calculate \( (1121)^{\frac{2}{3}} \). First, find the cube root of 1121 and then square the result.
Calculating the cube root of 1121:
\[
\sqrt[3]{1121} \approx 10.513
\]
Now squaring this result:
\[
x \approx (10.513)^2 \approx 110.56
\]
Finally, rounding to the nearest whole number gives:
\[
x \approx 111
\]
Thus, the average distance of Planet B from the center star is approximately **111 million kilometers**.
**Final Answer: 111 million kilometers**.