Kepler's 3rd law of planetary motion is exactly
what we need in order to answer this one:
(orbital period)² / (orbital radius)³ =
the same number for all bodies orbiting the sun
Let's call that number 'K' just for convenience.
So we know that T₀² / R₀³ = K
We're going to be looking for 'T', so let's rearrange the equation now.
Multiply each side by R₀³ .
Now it says
T₀² = K R₀³
Now, take the square root of each side, and we have
T₀ = √ (K R₀³) .
Now the radius is increased to (1.04 R₀).
We want to find the new T .
T = √ K · (1.04 R₀)³
= √ K · 1.124864 R₀³
Pull that decimal out of the radical, by taking its square root:
= 1.0606 √K · R₀³
T = 1.0606 T₀
The orbital time has increased by 6% .
_________________________________________
I suspect we probably could have said that since T varies
as R^1.5 power, we should look for (1.04)^1.5 power.
(1.04)^1.5 = 1.0606 <== bada-bing