Unit 5 Study Guide

Consider the function f(x)=p(0.5)^{x}+q, where p and q are constants. The graph of f(x) passes through the points (0,6) and (1,4).

The pH of a solution measures its acidity and can be determined using the formula pH=-log_{10}C, where Cis the concentration of hydronium ions in the solution, measured in moles per liter. A lower pH indicates a more acidic solution.

The population of mosquitoes decreases exponentially. The size of the population, P, after t days is modeled by the equation P=3200\cdot2^{-t}+50 where t\geq0.

Little Green island originally had no turtles. After 55 turtles were introduced to the island, their population is modeled by N(t)=a\cdot2^{-t}+10, t\geq0,where a is a constant and t is the time in years since the turtles were introduced.

A population of 200 rabbits was introduced to an island. One week later the number of rabbits was 210. The number of rabbits, N , can be modeled by the function N(t)=200×b^{t},  t\geq0 , where t is the time, in weeks, since the rabbits were introduced to the island.

Sejah placed a baking tin, that contained cake mix, in a preheated oven in order to bake a cake. The temperature in the center of the cake mix, T, in degrees Celsius (°C) is given by T(t)=150-a(1.1)^{-t}, where t is the time, in minutes, since the baking tin was placed in the oven. The temperature in the centre of the cake mix was 18 °C when placed in the oven.

The growing population of cats in my gnome garden forced the gnomes to migrate to my attic, so now the population of gnomes in my attic grows according to the logistic model P(t)=\frac{110}{1+27.3e^{-0.14t}}, where t is measured in days.

Data for a small city’s population is shown below. P(t)=\frac{500,000}{1+9e^{-0.125t}}, t\geq0 where t is the number of years since 1980.

A cup of boiling water is placed in a room to cool. The temperature of the room is 20°C . This situation can be modelled by the exponential function T=a+b(k^{-m}), where T is the temperature of the water, in °C , and m is the number of minutes for which the cup has been placed in the room. A sketch of the situation is given as follows.