Rates of change
1. A raindrop is considered to grow at a rate dependent by direct proportion upon its surface area. Its mass is dependent upon its volume. Find an expression for the rate of change of volume for a spherical raindrop.
2. A spherical crystal, radius r, grows at a constant change of volume, a. Find an expression for the rate of change of surface area in terms of r and a.
3. different crystal has projections which can be modelled as cones, excluding the base. If the cones are assumed to be sufficiently polygonal to be treated as having circular bases then, assuming that growth is directly proportional to surface area, dA/dt = µ say, explore how the volume changes. Start with a situation where the slant height, l, of a cone is expressed in terms of the radius, l=kr
4. An iceberg melts so that it loses volume in direct proportion to the submerged surface area. Assuming that, initially, a berg is cubic [of side r metres] and that the top face is parallel to and clear of any sea surface and that 90% of such a berg is submerged, find an expression for the short-term rate of change of volume, showing that this is a constant.
In practice, a berg will rotate when sufficient erosion has taken place below. Assume that the initial position has side R0. It is suggested that the melting berg could be considered to be an (upper) cuboid plus a (lower, smaller) cube of side s= r-R0/10. Rotation will occur at around the time when the residual subsurface volume is similar to the super-surface volume. Rotation occurs at or before the moment when s3= R03/10. Show that this may suggest a value of r=0.536R0 as the upper bound for the point of rotation.
The model is weak because it fails to include the increased surface area as undercutting occurs. This is approximately an additional area of R02 - s2. Show that the rate of change of area is now the previous constant dr/dt multiplied by 8s. So how does the rate of loss of volume now relate to s?
Note that the modelling rapidly becomes difficult; applied maths is difficult, as we so rarely use it to attempt real-world problems.
5. Variable x is changing at a rate in direct proportion to x, so that dx/dt=kx. Show that x = µekt is a solution to this differential equation.
If x is the side of a cube that is expanding in this way, show that the rate of change of area, A, and of volume, V, are also directly proportional to A and V respectively, and find the expansion rates in terms of k.
6. A population of size N is varying in a cyclical manner across t months and can be modelled as N=A sin(wt+c). If the initial population is N0 and briefly unchanging and if the period of the cycles is annual, then rewrite the model, trying to eliminate A, w and c. If also the minimum population is half of the maximum population, express the population size, N, entirely in terms of N0 and t. Using a sketch, explain how there are two possible solutions to this problem.
7. Write an equation to model the height of water in a tidal basin. Think of water height against a pier or jetty. Use units of metres and hours. As help, the time between tides is about 6.5 hours. Let y be the height, y0 the lowest tide level and y1 the high tide level.
If the tidal range changes in a repeating monthly cycle, how would you modify your model?
DJS 20130417