STATEMENT OF PROBLEM

Create a mathematical model to help answer the question:

(1) How long does it take an ant to build a tunnel?

That is, construct a mathematical model to find T(x) the time it will take an ant to build a tunnel x units long.

More precisely: How long does it take an ant to dig a straight tunnel of length x in uniform, moist, fine sand into the side of a vertical wall?

Use your model to answer the following question:

(2) If we double the length of the tunnel then what happens to the time it takes to dig the tunnel?

From observations about ants and a desire to emphasize the simplicity of the situation rather than the complexity involved one can safely make the following five assumptions:

(a) The ant will build a level straight tunnel.

(b) The ant will build in uniform, moist, fine sand.

(c) The ant will walk as fast when it is carrying something as when it is unburdened. It would take more energy to do so, but the pace is the same.

(d) The tunnel's cross-sectional area is constant.

(e) The ant is digging into the side of a sand wall and the entry hole is high above the ground at the foot of the wall.

This last assumption we throw in so that the issue of what to do with thesand when we get it to the mouth of the tunnel does not complicate the modeling activity too much.