**Calculation of Retardation and Time for Brakes Applied**

To calculate the retardation produced by the brakes and the time for which the brakes are applied, we can use the equations of motion. The given information includes the initial speed of the truck, the final speed of the truck, and the distance over which the truck slows down.

**Given data:**
- Initial speed (u) = 90 km/h
- Final speed (v) = 50 km/h
- Distance (s) = 20 m

**Conversion of Units:**
Before proceeding with the calculations, we need to convert the speeds from km/h to m/s, as the SI unit for speed is meters per second (m/s).

1 km/h = 1000 m/3600 s
Therefore, the initial speed (u) is converted as follows:
- u = 90 km/h = (90 * 1000) m/3600 s = 25 m/s

The final speed (v) is also converted:
- v = 50 km/h = (50 * 1000) m/3600 s = 13.89 m/s

**Calculating Retardation:**
The equation of motion that relates initial speed, final speed, retardation, and distance is given by:
v^2 = u^2 + 2as

Rearranging this equation, we can solve for retardation (a):
a = (v^2 - u^2) / (2s)

Substituting the values we have:
a = (13.89^2 - 25^2) / (2 * 20)

Simplifying the equation:
a = (-381.5621) / 40
a = -9.539 m/s^2

Therefore, the retardation produced by the brakes is approximately 9.539 m/s^2.

**Calculating Time for Brake Application:**
To calculate the time for which the brakes are applied, we can use the equation of motion relating initial speed, final speed, retardation, and time:
v = u + at

Rearranging this equation, we can solve for time (t):
t = (v - u) / a

Substituting the values we have:
t = (13.89 - 25) / (-9.539)

Simplifying the equation:
t = -11.11 / (-9.539)
t = 1.164 s

Therefore, the time for which the brakes are applied is approximately 1.164 seconds.

**Explanation:**
The retardation produced by the brakes is a measure of how quickly the truck slows down. In this case, the negative sign indicates that the retardation is in the opposite direction to the initial motion of the truck, as the truck is slowing down.

The time for which the brakes are applied is the duration during which the brakes exert a force on the truck to slow it down. It is calculated using the equation of motion and represents the time taken for the truck to decelerate from the initial speed to the final speed.

It is important to note that the calculations assume constant retardation throughout the slowing down process. In reality, factors such as friction, air resistance, and the condition of the brakes may cause the retardation to vary. These calculations provide an approximation based on the given data.