A bacterial culture in a petri dish obeys a logistic growth model
MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
A bacterial culture in a petri dish obeys a logistic growth model. A qualitative slope field for the governing differential equation is described as follows.
• On the time interval 0≤t≤12 hours and concentration interval 0≤y≤500 cells/mm2, all slope segments are horizontal at y=0 and at y=450.
• Along the line t=0, the slopes at y=50, y=225, and y=400 are approximately 8, 20, and 6, respectively.
• For 0<y<225 the slope magnitudes increase with y; for 225<y<450 the slope magnitudes decrease with y.
Suppose the culture begins with y(0)=50 cells/mm2.
(1) Based on the slope field, estimate the concentration at which the culture’s growth rate is maximal. Justify your estimate using the pattern of slope magnitudes.
(2) Determine whether the solution curve is concave up or concave down at t=2 hours and at t=8 hours, and provide an explanation tied directly to the qualitative features of the slope field.
(3) Assume a chemical introduced at t=6 hours instantly lowers the horizontal line of zero slopes from y=450 to y=350. Sketch qualitatively how the existing solution curve is affected for t>6, and explain the biological significance of this change.