You notice that your garden hose (with diameter of 5/8 in) fills a two-gallon bucket in about two minutes. (a) What is the flow velocity of the water exiting the hose?

flow velocity of the water leaving the end of the hose is 31.87 cm/s or 0.3187 m/s

Explanation:

Given that  

Diameter of hose, d = 5/8 in

changing to cm becomes

1 inch =2.54 cm

5/ 8 inch=0.625 x 2.54= 1.5875cm

Volume of water  filled  = 2gallons

changing  to cubic centimeters

1 gallon =  3785.41cm^3

2 gallons = 3785.41 x 2 =7570.82cm^3

Time, t = 2 min

60 sec= 1 min

?? sec = 2 min

= 120 secs

The volumetric flow rate of water, F is given as

F = V/t = 7570.82cm^3÷ 120 seconds

F = 63.09 cm^3/s

We Know that The volumetric flow rate is also equal the cross sectional area of pipe times  the speed of flow(velocity of flow). ie  

F = Av

v = F/A

Since Area A = πd^2/4

 v = F/(πd^2/4)

v = 4F/πd^2

Puting the given values;

v = (4× 63.09 )/(π×1.5875²)

v = 252.36/πx 2.5202

v = 31.869 cm/s =31.87cm/s 0r 0.3187m/s

Therefore flow velocity of the water leaving the end of the hose is 31.87 cm/s or 0.3187 m/s