Lake Champlain, bordered by New York, Vermont, and Quebec, has a surface area of 1,126 km^2. If the energy (173 PJ) is provided by sunlight shining on the lake at an average rate of 200 W/m^2 but with only half that amount absorbed, how long will it take the lake to absorb the energy needed to melt it?

how thick is the ice?

To find the time it takes for the lake to absorb the energy needed to melt it, we need to calculate the total energy absorbed by the lake and then divide it by the average rate of energy absorption.

First, let's find the total energy absorbed by the lake. We know that the lake's surface area is 1,126 km^2, which is equal to 1,126,000,000 m^2 (since there are 1,000,000 m^2 in 1 km^2).

The average rate of energy absorption is given as 200 W/m^2, but only half of that amount is absorbed.

To find the total energy absorbed, we multiply the lake's surface area by the energy absorption rate per square meter and then multiply it by 0.5 (since only half is absorbed).

Total absorbed energy = Surface area * Energy absorption rate * 0.5
= 1,126,000,000 m^2 * 200 W/m^2 * 0.5

Now, we can calculate the total absorbed energy:

Total absorbed energy = 1,126,000,000 m^2 * 200 W/m^2 * 0.5
= 112,600,000,000 W

Next, we need to convert the total absorbed energy from watts (W) to joules (J) by multiplying it by the conversion factor of 1 J = 1 W * 1 s (since 1 watt is equal to 1 joule per second):

Total absorbed energy = 112,600,000,000 W * (1 J / 1 W * 1 s)
= 112,600,000,000 J

Finally, we can calculate the time it takes for the lake to absorb the energy needed to melt it by dividing the total absorbed energy by the average rate of energy absorption:

Time taken = Total absorbed energy / Average rate of absorption
= 112,600,000,000 J / 173 PJ

Before dividing, let's convert petajoules (PJ) to joules (J) using the conversion factor 1 PJ = 1,000,000,000,000,000 J:

Time taken = 112,600,000,000 J / (173 * 1,000,000,000,000,000 J)
= 112,600,000,000 / 173,000,000,000,000

Finally, calculate the time taken:

Time taken = 0.0006506 seconds

Therefore, it will take approximately 0.0006506 seconds for the lake to absorb the energy needed to melt it.