I think this is supposed to be an autonomous linear differential equation...t is not supposed to appear because the rate of change of the height of snow only depends on the amount of snow on the ground. It is equal to the difference between the rate of snowfall and the rate of snow removal, but the rate of snowfall is a constant...why is that? I'm having troubling understanding exactly why only 3/5 and not 3/5t appears in the differential equation
I think this is supposed to be an autonomous linear differential equation...t is not supposed to appear because the rate of change of the height of snow only depends on the amount of snow on the ground. It is equal to the difference between the rate of snowfall and the rate of snow removal, but the rate of snowfall is a constant...why is that? I'm having troubling understanding exactly why only 3/5 and not 3/5t appears in the differential equation
I think this is supposed to be an autonomous linear differential equation...t is not supposed to appear because the rate of change of the height of snow only depends on the amount of snow on the ground. It is equal to the difference between the rate of snowfall and the rate of snow removal, but the rate of snowfall is a constant...why is that? I'm having troubling understanding exactly why only 3/5 and not 3/5t appears in the differential equation
I think this is supposed to be an autonomous linear differential equation...t is not supposed to appear because the rate of change of the height of snow only depends on the amount of snow on the ground. It is equal to the difference between the rate of snowfall and the rate of snow removal, but the rate of snowfall is a constant...why is that? I'm having troubling understanding exactly why only 3/5 and not 3/5t appears in the differential equation
With integration, one of the major concepts of calculus. Differentiation is the derivative or rate of change of a function with respect to the independent variable.
Expert Solution
Step 1: Solution
You're correct in thinking that this situation can be modeled by an autonomous linear differential equation.
The reason the rate of snowfall appears as a constant (3/5) and not as a function of time (3/5t) in the differential equation is because the problem statement specifies that snow is falling at a constant rate of 3/5 inches per hour. This means that the rate of snowfall is not changing over time, it remains constant throughout the entire process.
Let's denote the height of the snow at time t as h(t). According to the problem statement, the rate of change of the height of the snow, dh/dt, is equal to the difference between the rate of snowfall and the rate of snow removal.
