1. Question: Consider two planets with uniform mass distributions. The mass density and the radius of planet 1 are p1 and R1, respectively, and those of planet 2 are p2 and R2.
What is the ratio of their masses?
Ans: M1/M2 = (p1/p2)(R1/R2)^3...I got this answer correct.
I need help in the next, as below, affiliated question.
2. What is the ratio of the circular areas defined by the two equators? Can you please tell me which one you think is correct?
1. A1/A2 = (R2/R1)^3
2. A1/A2 = (R2/R1)^2
3. A1/A2 = pi (R1/R2)^3
4. A1/A2 = (R2/R1)
5. A1/A2 = (R1/R2)^3
6. A1/A2 = pi (R2/R1)^2
7. A1/A2 = (R1/R2)
8. A1/A2 = (R1/R2)^2
9. A1/A2 = pi (R2/R1)^3
10. A1/A2 = pi (R1/R2)^2
What I am thinking is that in the top question, R refers to the radius, so R should also stand for the radius in the below question since they are both connected. But the bottom question talks about "equators" which I take to be diameters. That is why I am confused; if the bottom question's R's are radii, then only those with an exponent of 2 and pi should be right because the area of a circle is pi*r^2. So it should only be either 6 or 10. But if the bottom question has R as the diameter, then I don't know which one it is because the formula would be (pi*d^2)/4 (according to what I found online). Could you please tell me what you think it is, because I don't understand it? I also don't get how to tell whether R1 or R2 should be on the top.
The answer is
A1/A2=(R1/R2)^2
Not exactly sure how to explain but i'll give it a shot.
Since the area is already first over the 2nd, the radius should also be the same. So pluggin in the radius, it would also be r1/r2 to match the a1/a2 part. Not sure but somehow the pis just cancel out. I originally picked A1/a2=(r1/r2) but got that incorrect and chose the ^2 one. Got it right. Can't really explain it. I sort of did half math and half instinct.
To determine the correct ratio of the circular areas defined by the two equators, let's analyze the problem.
First, let's establish some definitions:
- R1 and R2 represent the radii of planet 1 and planet 2, respectively.
- A1 and A2 represent the circular areas defined by the equators of planet 1 and planet 2, respectively.
Now let's address your confusion about the use of "R" in the question. In both the top and bottom questions, "R" refers to the radius of the respective planets. In the top question, it is used to calculate the ratio of their masses, while in the bottom question, it will be used to calculate the ratio of their circular areas.
To find the ratio of the circular areas, we need to consider the formula for the area of a circle, which is A = π * r^2, where "r" is the radius of the circle.
To proceed, we need to determine whether the "R" mentioned in the bottom question represents the radius or the diameter of the equator.
If we assume that "R" in the bottom question represents the radius, then the correct formula to calculate the ratio of the circular areas would be A1/A2 = (R2/R1)^2 (option 8). This is because the area of a circle is proportional to the square of its radius.
However, if we assume that "R" in the bottom question represents the diameter, we need to convert the diameter to radius before using it in the area formula. In this case, the formula would become A1/A2 = π * (R2/2)^2 / (π * (R1/2)^2), which simplifies to A1/A2 = (R2/R1)^2. This means that option 8 would still be the correct answer.
Therefore, the correct ratio of the circular areas defined by the equators of the two planets is A1/A2 = (R2/R1)^2 (option 8) in both cases, whether "R" represents the radius or the diameter.
To find the ratio of the circular areas defined by the two equators, we need to consider the formula for the area of a circle, which is A = πr^2, where A is the area and r is the radius of the circle.
In this case, "R" refers to the radius of the planets, not the diameters as you mentioned. So we can eliminate options 4, 6, 7, and 9.
Since we are comparing the areas A1 and A2, we need to consider the ratio A1/A2.
The correct formula for the ratio of the circular areas defined by the two equators is:
Option 2: A1/A2 = (R2/R1)^2
This is because the radius squared (R^2) is directly proportional to the area of the circle. So, when comparing two areas, we compare the ratio of the radii squared. The squared term accounts for the relationship between the radius and the area.
In this formula, R1 should be on the bottom since we want to compare the area of the first planet's equator (A1) to the second planet's equator (A2). So, the correct answer is option 2.
I hope this helps clarify your confusion! Let me know if you have any further questions.