**Analysis of the Problem:**
To solve the problem, we need to determine the hydraulic conductivity of the aquifer and the transmissivity. The given information includes the diameter of the well, the drawdown at two different distances from the well, and the pumping rate.

**Definition of Terms:**
1. Diameter of the well: 20 cm
2. Drawdown at 10 m from the well: 3 m
3. Drawdown at 100 m from the well: 0.5 m
4. Pumping rate: 2720 litres per minute

**Darcy's Law:**
Darcy's law is used to determine the flow of water through a porous medium. It states that the flow rate is directly proportional to the hydraulic conductivity and the hydraulic gradient, and inversely proportional to the cross-sectional area.

**Calculating the Hydraulic Gradient:**
The hydraulic gradient can be calculated by taking the difference in drawdown and dividing it by the distance between the two points.

Hydraulic Gradient = (Drawdown at 10 m - Drawdown at 100 m) / (Distance between the two points)

Hydraulic Gradient = (3 m - 0.5 m) / (100 m - 10 m) = 2.5 m / 90 m = 0.0278

**Calculating the Cross-Sectional Area:**
The cross-sectional area can be calculated using the formula for the area of a circle.

Cross-Sectional Area = π * (Radius of the well)^2

Radius of the well = Diameter of the well / 2 = 20 cm / 2 = 10 cm = 0.1 m

Cross-Sectional Area = π * (0.1 m)^2 = 0.0314 m^2

**Calculating the Flow Rate:**
Using Darcy's law, the flow rate can be calculated as follows:

Flow Rate = Hydraulic Conductivity * Cross-Sectional Area * Hydraulic Gradient

Flow Rate = Hydraulic Conductivity * 0.0314 m^2 * 0.0278

Flow Rate = Hydraulic Conductivity * 0.0008729 m^3/s

**Converting the Pumping Rate:**
The pumping rate is given in liters per minute. To convert it to cubic meters per second, we need to divide it by 1000.

Pumping Rate = 2720 liters per minute = 2720 / 1000 m^3/s = 2.72 m^3/s

**Calculating the Hydraulic Conductivity:**
By equating the flow rate from Darcy's law and the pumping rate, we can solve for the hydraulic conductivity.

Hydraulic Conductivity * 0.0008729 m^3/s = 2.72 m^3/s

Hydraulic Conductivity = 2.72 m^3/s / 0.0008729 m^3/s = 3116.81 m/s

**Calculating the Transmissivity:**
The transmissivity can be calculated by multiplying the hydraulic conductivity by the thickness of the aquifer.

Transmissivity = Hydraulic Conductivity * Thickness of the Aquifer

As the thickness of the aquifer is not given in the problem, we cannot calculate the transmissivity without this information.

In conclusion, we have determined the hydraulic conductivity of the aquifer to be 3116.81 m/s using