Consider an asset that costs $670,000 and is depreciated straight-line to zero over its eight-year tax life. The asset is to be used in a five-year project; at the end of the project, the asset can be sold for $95,000. If the relevant tax rate is 35 percent, the aftertax cash flow from the sale of this asset is $

I am having a very hard time figuring out how to do this problem. I can not find in my text anything similar to this question. How would I go about doing this?

For what it's worth, here's how I would approach it.
For a $670,000 asset, straight-line over 8 years, you would have $83,750 per year depreciation. At a 35% tax rate, that is $29,312.50 each year cash flow contribution.

Selling at the end of year 5, you have $251,250 remaining on the books. Collecting $95,000 on the sale leaves you with a net loss of $156,250. At 35% that gives you an after tax contribution of $54,687.50. You also have the $95,000 in your pocket.

If it is truly a 5 year project and the asset is sold at the end of it, that would mean to me that the sale took place very early in year 6.

So, for after tax cash flow, I would get:
Year Cash Flow
1 $29,312.50
2 $29,312.50
3 $29,312.50
4 $29,312.50
5 $29,312.50
6 $149,687.50

If you assume the sale was completed in year 5, you'd add my year 6 numbers to the year 5 numbers.

This assumes that the usage years match your fiscal year. If they don't there would be proration in years 1 and 6 driven by the difference.