The equation states that In (10w) = 4.6 + 3lnh, so we can rewrite it as ln (10w) = 4.6 + 3lnh.
We can then use the property of logarithms that states that ln (a^b) = b ln a to rewrite the equation as ln (10^(10w)) = 4.6 + 3lnh.
This simplifies to 10w = 4.6 + 3lnh, or w = (4.6 + 3lnh)/10.
Substituting the given value of w = 60, we get 60 = (4.6 + 3lnh)/10.
Solving for h, we find that h = e^((60 - 4.6)/3).
To find the value of h, we need to use the given information about the logarithms of 2, 3, and 5. We know that In 2 = 0.6, In 3 = 1.1, and In 5 = 1.6, so we can rewrite the expression for h as:
h = e^((60 - 4.6)/3) = e^(55.4/3) = e^(18.466666666666665)
We can use the fact that e^(In x) = x to rewrite this expression as:
h = 2^(18.466666666666665/0.6) = 3^(18.466666666666665/1.1) = 5^(18.466666666666665/1.6)
We can then use the given values of In 2, In 3, and In 5 to find that h = 2^30.777 = 3^16.727 = 5^11.541.
Since the height h is in meters, the final answer is h = 30.777 meters.