Albedo equations giving different results

[Sabrewolf]

1. This question isn't so much a homework question per se, however I am having difficulty using certain equations to find the albedo of the Earth. In attempting to find the albedo, I am using two equations, shown below, however each equation gives a different answer. I am given the following variables:

- Power absorbed at the surface of the Earth is 240 W/m^2
- Solar Constant(S) is 1.37 kW/m^2
- Earth is assumed to be a perfect blackbody, thus emissivity is 1
- Temperature(T) on the Earth is assumed to be 255 kelvin
σ is Stefan Boltzmann constant, 5.67 x 10^-8
ε is emissivity of the Earth
α is albedo

Homework Equations

T = $$\sqrt[4]{\frac{S(1-\alpha)}{4\epsilon\sigma}}$$ <-- this is a given equation found by combining the Stefan Boltzmann law with an equation for incoming power.

albedo = total scattered power/total incident power <--this is from a provided data booklet

The Attempt at a Solution

The actual work isn't too hard, when the 1st equation is moved around to find the albedo, it reads:

$$\alpha$$ = -$$\frac{T^{4}4\sigma}{S}$$ + 1

This equation, when solved, gives the answer $$\alpha$$=0.3, this is the correct response according to the book

When I use the second equation, I take the solar constant S as the total incident power. In order to find the total scattered power, I'm subtracting the amount absorbed (given as 240W/$$m^{2}$$) from the solar constant because the difference isn't absorbed by the earth, meaning it is reflected into space or scattered. However I get this:

[$$\frac{1.37*10^{3} - 240}{1.37*10^{3}}$$ = 0.82

This value is incorrect and I'm not sure why, given a presumed correct equation, I'm getting a wrong answer. Am I misusing the equation somehow?

 

[haruspex]

The given data are inconsistent. To match the rest of the data, the power absorbed should be 410W/m². Maybe the 240 is a typo for 420.