The time taken by Mars to revolve round the sun is 1.88 years. Find the ratio of average distance between mars and the sun to that between the earth and the sun.

Explanation

To solve the problem of finding the ratio of the average distance between Mars and the Sun to that between the Earth and the Sun, we will use Kepler's Third Law of planetary motion. Here’s a step-by-step solution:

### Step 1: Understand Kepler's Third Law
Kepler's Third Law states that the square of the time period (T) of a planet's orbit is directly proportional to the cube of the semi-major axis (r) of its orbit. Mathematically, it can be expressed as:
\[ T^2 \propto r^3 \]
This can be rewritten for two planets (Earth and Mars) as:
\[ \frac{T_m^2}{T_e^2} = \frac{r_m^3}{r_e^3} \]
where \(T_m\) and \(T_e\) are the time periods of Mars and Earth, and \(r_m\) and \(r_e\) are their respective average distances from the Sun.

### Step 2: Assign Known Values
From the problem, we know:
- The time period of Mars, \(T_m = 1.88\) years.
- The time period of Earth, \(T_e = 1\) year.
- Let the average distance between Mars and the Sun be \(r_m\) and the average distance between Earth and the Sun be \(r_e\).

### Step 3: Set Up the Ratio
Using Kepler's Third Law, we can set up the equation:
\[ \frac{T_m^2}{T_e^2} = \frac{r_m^3}{r_e^3} \]
Rearranging gives us:
\[ \frac{r_m^3}{r_e^3} = \frac{T_m^2}{T_e^2} \]

### Step 4: Substitute the Known Values
Substituting the known values into the equation:
\[ \frac{r_m^3}{r_e^3} = \frac{(1.88)^2}{(1)^2} \]
Calculating \( (1.88)^2 \):
\[ (1.88)^2 = 3.5344 \]
Thus, we have:
\[ \frac{r_m^3}{r_e^3} = 3.5344 \]

### Step 5: Find the Ratio of Distances
To find the ratio of the distances, we take the cube root:
\[ \frac{r_m}{r_e} = \left(3.5344\right)^{1/3} \]
Calculating the cube root:
\[ \frac{r_m}{r_e} \approx 1.52 \]

### Final Answer
The ratio of the average distance between Mars and the Sun to that between the Earth and the Sun is approximately:
\[ \frac{r_m}{r_e} \approx 1.52 \]

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