Question 1: For a rectangular lawn, we can establish a mathematical model of fixed sprinkler layout to make the water spraying as uniform as possible and waste as little water as possible. Specifically, we can determine the best position of the sprinkler by solving the minimum coverage circle problem, so that every point of the lawn is covered by at least one sprinkler. In this way, the sprayed water can be distributed as evenly as possible on the whole lawn.

Question 2: After determining the layout of sprinklers, we need to determine the frequency of watering to keep the lawn at a reasonable humidity level. What we need to pay attention to is that the rate of water supply during irrigation cannot be greater than the absorption and permeability of soil, otherwise water will overflow around the ground and cause waste. Therefore, we need to determine the frequency of water spraying according to the absorbability and permeability of soil. Specifically, we can determine the total water supply capacity of the irrigation system according to the lawn area, the number of sprinklers and the spraying rate of sprinklers. Then, we can determine the time required for each irrigation according to the absorbability and permeability of the soil. For example, the absorption permeability of soil is per square meter per hour 1mm, and the spraying rate of sprinkler is 0.5mm per square meter per hour, so the time required for each irrigation is 2 hours.

Question 3: Finally, the mathematical model needs to be adjusted and optimized in combination with the 24-hour weather forecast. Specifically, we need to consider the influence of weather factors such as rainfall, temperature and sunshine time on lawn water demand. For example, when the rainfall is heavy, the irrigation frequency can be reduced; When the temperature is high and the sunshine time is long, it is necessary to increase the irrigation frequency. In addition, we can also predict the future water demand according to the weather forecast, so as to better arrange the irrigation plan.

To sum up, we can optimize the lawn irrigation system by establishing a mathematical model to make the water spraying as uniform as possible and reduce waste, and at the same time consider the influence of weather factors on lawn water demand. These mathematical models can help us manage lawn irrigation system better, thus saving water resources and protecting the environment.