Exponential and Logarithmic Models: Apply It

  • Model exponential growth and decay.
  • Use Newton’s Law of Cooling.
  • Use logistic-growth models.
The general form for an exponential growth model is [latex]A=A_0e^{rt}[/latex]. (Growth if [latex]r>0[/latex], decay if [latex]r<0[/latex].)

A population is measured in regular time steps:

[latex]t[/latex] [latex]A(t)[/latex]
0 500
5 402
10 323
15 260

Write the exponential model.

The general form for Newton’s Law of Cooling:[latex]T(t)=A e^{kt}+T_s[/latex], where [latex]T_s[/latex] is the ambient temperature (horizontal asymptote), [latex]A=T_0-T_s[/latex], and [latex]k[/latex] is found from any additional point.
On a cool morning, a mug of tea is set on a patio where the air temperature is 10°C. The temperature of the tea is shown in the graph:Coordinate plane with a horizontal red dotted line that cross the y-axis at 10, and a black curved arc the cross the y-axis at 80 and continues down towards the red line as the slope gets less steep. Green dot at the point (20,30) on the black curve.Write the cooling model.