Nest lookup Function to lookup name between date range and return value

Hi, I am struggling to write a lookup function that will look at column a and b in the inv tab and return the rate in column c that meets that criteria from the cost page. I have the lookup between range built out but cant seem to get the lookup for the name.

Thank you for the help!

Your output is wrong

Inv!C2 should be 173 not 159

This works
in Inv!C2
=SUMPRODUCT((A2=Cost!A$2:A$1000)*(B2>=Cost!B$2:B$1000)*(B2<=Cost!C$2:C$1000)*(Cost!D$2:D$1000))
and copy down the column