Question

Say a dime is held 100 cm from the observer and appears to have the same size (this is called having the same angular diameters). If the planet is 4 x 10⁵ km away, what is the radius of the planet? How does the planet compare in volume to the Earth?

Answer 1

We can use the formula

alpha = 57.3*d/e ------------ (1)

where d is the actial diameter of coil and e is the distance between observer and coin which is given as 100 cm and alpha is angular diameter (You have not provided diameter of coin)

From this we can find the value of alpha.

Now as it is given that these have the same angular diameter.

so, alpha = 57.3 x D/E where D is diameter of planet and E is distance to planet given as 4x 10⁵ km

using the alpha from (1), calculate D,

D = alpha x E / 57.3

after calculating D, divide by 2 to get the radius.