When the planet Jupiter is at a distance of 824.7 million kilometres from the earth, its angular diameter is measured to be 35.72 s of arc. Calculate the diameter of Jupiter.

The correct answer/option is 1.428x10⁵ km.

Step-by-Step Explanation:

1. Identify the given values:
Distance of Jupiter from Earth, $D=824.7\text{million km}=824.7\times {10}^6\text{km}$
Angular diameter of Jupiter, $\theta ={35.72}^{″}$ (seconds of arc)

2. Convert the angular diameter from seconds of arc to radians:
We know that:
$1^{″}=\frac{\pi }{180\times 3600}\text{radians}\approx 4.85\times {10}^{-6}\text{rad}$
Therefore, the angular diameter in radians is:
$\theta =35.72\times 4.85\times {10}^{-6}\text{rad}$
$\theta \approx 1.732\times {10}^{-4}\text{rad}$

3. Calculate the linear diameter of Jupiter:
Let $d$ be the diameter of Jupiter. The relation between the distance $D$, angular diameter $\theta$, and linear diameter $d$ is given by:
$d=D\times \theta$
Substitute the values into the formula:
$d=(824.7\times {10}^6\text{km})\times (1.732\times {10}^{-4}\text{rad})$
$d\approx 1428.38\times {10}^2\text{km}$
$d\approx 1.428\times {10}^5\text{km}$

Thus, the diameter of Jupiter is approximately 1.428x10⁵ km.