Question 810511: Hello! My name is Nevan and I am currently in Alg 2. I have this assignment due in a few days and I got stuck on this problem. I would love a answer and explanation so I know how to do in the future. Much appreciated!
-----------------------------Problem Below----------------------------
Cockroaches are pretty large bugs that breed very quickly.
For the purposes of this assignment, we will assume the following:
Cockroaches breed quickly enough that their population doubles every minute.
It takes 10 cockroaches to cover one square inch of the ground.
Assume that you start with a population of 1 cockroach (don't ask how it reproduces). Then the cockroach population is p=2t, where t is measured in minutes.
Time = t (In Minutes)
Population of Cockroaches = p
144 sq. inch in 1 sq. foot
(I answered 1 and 2 already)
#1 isn't related to the cockroaches problem. But here it is.
1) If the proportionality constant for the birth rate is greater than that for the death rate, are you going to have exponential growth or decay? Explain.
2) How many Cockroaches cover 1 sq. inch?
3) How long until there are enough to cover the floor of an 8' by 12' kitchen (make sure to convert from feet to inches)?
4) How long until they cover the floor of a typical 2000 square foot house?
5) How long until they cover an area the size of the city of Colorado Springs (about 186 square miles)?
6) How long until they cover an area the size of Colorado (104, 185 square miles)?
7) How about an area the size of the United States (3,539,289 square miles including all 50 states plus DC)?
8) How about covering the world (57,900,000 square miles)?
Answer by rothauserc(4718)   (Show Source):
You can put this solution on YOUR website! 3) 8' = 12*8 = 96 inches
12' = 12*12 = 144 inches
Area = 96 * 144 = 13824 square inches
p = 13824 * 10 = 138240 cockroaches
now we know that the population doubles each minute, we start with 1 cockroach, after 1 minute we have 2, after 2 minutes we have 4, after 3 minutes we have 8
so we use p = 2^t to give us the population after t minutes
138240 = 2^t
we use the definition of logarithm to calculate t
sooooooo here we go!
t = log base 2 (138240)
t = 17.08 minutes
4) 2000 * 144 = 288000 square inches
p = 288000 * 10 = 2880000
t = log base 2 (2880000) = 21.46 minutes
5) 186 square miles = 27878400 square feet * 186 square miles * 144 square inches = 7.466950656e9 square inches
p = 7.466950656e9 * 10 = 7.466950656e10
t = log base 2 (7.466950656e10) = 36.12 minutes
6) 104185 square miles = 27878400 square feet * 104185 square miles * 144 square inches = 4.18249598976e14 square inches
p = 4.18249598976e14 * 10 = 4.18249598976e15
t = log base 2 (4.18249598976e15) = 51.89 minutes
7) 3539289 square miles = 27878400 square feet * 3539289 square miles * 144 square inches = 1.42084388818944e16 square inches
p = 1.42084388818944e16 * 10 = 1.42084388818944e17
t = log base 2 (1.42084388818944e17) = 56.98 minutes
8) 57900000 square miles = 27878400 square feet * 57900000 square miles * 144 square inches = 2.3243894784e17 square inches
p = 2.3243894784e17 * 10 = 2.3243894784e18
t = log base 2 (2.3243894784e18) = 61.01 minutes
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