Final answer:
A formula to calculate the mass m(t) of a rolling, snow-accumulating cylinder is based on the rate at which it accumulates snow over its surface area. The differential equation linking the rate of change of mass and radius, dr/dt, can be obtained from the relationship between mass, density, and volume of the cylinder.
Step-by-step explanation:
To answer the student's question regarding a cylinder of snow accumulating mass as it rolls downhill, we first need to provide a formula for the mass m(t) of the cylinder. Since the snow is accumulating on the curved surface area of the cylinder, and the rate of accumulation is directly proportional to this area, we can express the rate of mass gain as dm/dt = B × (2πr(t)L), where B is the proportional constant, r(t) is the radius of the cylinder, and L is its fixed length.
To find the rate of change of the radius dr/dt, we need to understand the relationship between the mass and the volume of the cylinder. The mass m(t) is equal to the density ρ times the volume V of the cylinder, so m(t) = ρ × (V) = ρ × (πr(t)^2L). Differentiating this equation with respect to time provides us with a method to express dr/dt.
The dependence of dr/dt on time t can be derived from this relationship, showing how the radius of the cylinder changes over time as it accumulates more mass.