Acircular backyard pool has a diameter of 26 feet and is 4 feet deep. one cubic foot of water has a capacity of approximately 7.48 gallons. if water costs $3 per thousand gallons, how much, to the nearest dollar, will it cost to fill the pool?

Find the water volume of the pool.  The formula for a vertical cylinder is V = pi*r^2*h, which here is V = pi(13 ft)^2*(4 ft) = 676*pi cubic feet.

Now convert this 676 cu. ft. to gallons:

Mult (676*pi cu ft) by the conversion factor (7.48 gal) / (1 cu ft):

676*pi cu ft     7.48 gal
* = 5056*pi gallons (rounded down from 5056.48 pi)
        1             1 cu ft

The cost of filling the pool will be

5056.48*pi gallons        $3
* = $48 (to the nearest dollar)
        1                           1000 gal