query to find closest lesser date

I have a table with rates. each rate has an effective starting date. (the ending date of any rate is implied by the existence of a record with a newer effective starting date.)
I also have a table with activities. Each activity occurs on a date.

What is the best way to match an activity to the rate that was effective on that date?

(unfortunately I don't have the ability to modify the data structure, so adding an explicit ending date for a rate is out.)

rates:
StartDate Rate
9/1/2010    17.00
10/1/2010   18.70
11/1/2010   20.00

Activities:
WorkCenter  ActionDate Hours
WC1         9/30/2010   10
WC1         10/1/2010   5
WC2         10/30/2010  8
WC2         11/3/2010   9

Desired result:
Workcenter ActionDate   Hours   Rate    Cost(=rate*hours)
WC1         9/30/2010   10      17.00   170.00
WC1         10/1/2010   5       18.70    93.50
WC2         10/30/2010  8       18.70   149.60
WC2         11/3/2010   9       20.00   180.00

SELECT 
    a.WorkCenter
  , a.ActionDate
  , a.Hours
  , r.Rate
  , r.Rate * a.Hours  AS Cost
FROM 
    Activities AS a
  JOIN
    Rates AS r
      ON  r.StartDate = 
          ( SELECT MAX(StartDate)
             FROM Rates 
             WHERE StartDate <= a.ActionDate
          ) ;

or a GROUP BY solution:

SELECT 
    a.WorkCenter
  , a.ActionDate
  , a.Hours
  , r.Rate
  , r.Rate * a.Hours  AS Cost
FROM 
    Activities AS a
  JOIN
    ( SELECT 
          a.ActionDate
        , MAX(r.StartDate) AS StartDate
      FROM 
          Activities AS a
        JOIN
          Rates AS r
            ON r.StartDate <= a.ActionDate
      GROUP BY a.ActionDate
    ) AS grp
      ON  grp.ActionDate = a.ActionDate
 JOIN
    Rates AS r
      ON  r.StartDate = grp.StartDate ;

StackExchange Database Administrators Q#27823, answer score: 12