Questions-

Question 1) The type of functional response of an organism gathering energy (e.g. a predator hunting and eating prey) dictates the rate of capture, r (prey per hour), as a function of the density of prey, d (hundred organisms per square mile). A model of this relationship for type III functional response is given by

r(d) =  ad²/1 + apd²,

where a is the attack rate (the speed at which a predator locates prey). p is the processing time (the time required for a predator to kill and digest prey). Assume for population Alpha that a = 0.2 and p = 0.5, while for population Beta a = 1.2 and p = 0.5, and for population Gamma a = 1.2 and p = 0.3.

a) Without choosing values for the constants, what is the long-term behavior of r? What does this imply about capture rate?

b) Rom the predators' perspective, should the value of a be large or small? What about the value of p?

c) What is the capture rate of population Alpha when the prey density is 200 per square mile? Find the capture rate at the same density for population Beta.

d) What prey density is required in order that population Beta have a capture rate of 1 per hour? Find the prey density required for population Gamma.

e) Compare the graphs of capture rate for populations Alpha and Beta on the interval (0, 15]. Justify the differences by examining the values of constants a and p in the context of predation.

f) Compare the graphs of capture rate for populations Beta and Gamma on the interval [0, 15]. Justify the differences by examining the values of constants a and p in the context of predation.

Question 2) Consider the following model for preventing the spread of a disease: A researcher believes that if x thousand individuals among a susceptible population are inoculated (made no longer susceptible to a disease), then a function of the form I(x) = a · R^x - b, for constants a, b, and R would model the eventual number of infected individuals (also in thousands).

a) If the susceptible population is 5000 and no one is inoculated, how many will eventually be sick? What point does that imply should be on the graph of the function I(x)?

b) If the susceptible population is 5000 and everyone is inoculated, how many will eventually be sick? What point does that imply should be on the graph of the function I(x)?

c) Assume that R = .9. Use the results of parts (a) and (b) to find approximate values for the constants a and b in the researcher's model.

d) According to the model, how many susceptible individuals will be infected if half of them are inoculated?

3) Suppose you put $500 in a bank account that pays 3.5% interest.

a) If the interest is compounded mutually, write a formula for the function B(t) that gives your balance in the account after t years. Compute the percent change of B(t) on the interval [n, n + 1). What do you notice?

b) If the interest is compounded monthly, write a formula for the function B(t) that gives your balance in the account after t years. What is the APY of this account? Again compute the percent change of B(t) on the interval [n, n+1]. What do you notice?

Question 4) If you put 81000 dollars in a bank account with an APY of 5.6% and Bill Gates puts $1.000,000,000 into a bank account that has an APY of 2.1%, when will you have more money than Bill Gates? Also, how much money would you have at that point in time?

Question 5) The decibel level of a sound is given by dB = 10 log (E/E₀), where E₀ = 10^-12 watts/m² and E is the intensity of the sound.

a) What is the decibel level of a sound with intensity E =10^-4?

b) Find the intensity of a 100 decibel sound.

c) How many times more intense is a 110 decibel sound compared to a 70 decibel sound?

d) The loudness (perceived volume by the human ear) of a sound of d decibels doubles each time the intensity of a sound increases by a factor of ten. To the human ear, how many times louder does a 110-decibel sound seem than a 70-decibel sound?