Fortunately, there is a simple formula for the power required to climb hills. It is quite accurate for speeds low compared with 10mph. At higher speeds, wind resistance and rolling resistance become significant.

So if you + bike weigh 200lb, and you are doing 10 mph up a 1/10 hill you get

You are also very fit! Most of us are not able to climb that fast for long. The 2002 Etape generally has gradients of 6%. Suppose you weigh 200lb with your bike, and are able to produce 150watts,

we get:

150 = 400 x speed x 0.06,

Note 1
In reality the power used to climb a hill is the sum of the power used for climbing (from the formula) + Wind resistance + rolling resistance.
But at climbing speeds of around 7mph wind resistance + rolling resistance only account for 20 Watts or so, compared with the more than 150 Watts needed to scale the hill. Wind resistence and rolling resistance are therefore fairly negligible, and may be ignored for practical purposes when climbing at relatively low speeds.

Note 2 The formula was derived from the fact that Power = Distance x Force per unit time.
Shuffling this you get Force x distance per unit time, or Force x speed.

The Force required to opose the gravitational vector is proportional to the gradient.
So to climb a hill, the power required is Force x Speed x Gradient. The formula uses an improbable mixture of units convenient units. The factor 2 at the front of the formula seems very unlikely. However if you start with foot pounds per second, shuffle this to give lb and feet per second, then convert to mph and watts, then multiply all the conversion factors together, you get a number so close to 2 that it make no difference! (less than 1% error!)