The exponential function modeled represents the number of square kilometers of land occupied by cane toads x years after this animal was first introduced into australia. based on the data, which measurement is closest to the number of square kilometers of land that will be occupied by cane toads 40 years after this animal was first introduced into australia?
a. 550,000 km2
b. 1,250,000 km2
c. 600,000 km2
d. 800,000 km2
part a
answer: a reasonable domain is where n is a real number. so n can be between 0 and 10. both endpoints are included.
work shown:
n is the number of days after the experiment starts. the smallest n can be is n = 0 which means that 0 days have gone by, and we're at the start. to find out how large n should be, then replace f(n) with 16.13 and solve for n. use logarithms to isolate the exponent.
f(n) = 12*(1.03)^n
16.13 = 12*(1.03)^n
16.13/12 = (1.03)^n
1.34416666666667 = (1.03)^n
(1.03)^n = 1.34416666666667
log[ (1.03)^n ] = log[ 1.34416666666667 ]
n*log[ 1.03 ] = log[ 1.34416666666667 ]
n = log[ 1.34416666666667 ]/log[ 1.03 ]
n = 10.0062999823929
this rounds to n = 10 which is fairly close but not 100% perfect. so this is the largest n can be.
note: if you plug n = 10 into f(n), you'll get roughly 16.127 which rounds to 16.13 (this comes up again in part c)
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part b
answer: the y intercept is 12. it represents the starting height of the plant in cm.
work shown:
plug n = 0 into the f(n) function. simplify
f(n) = 12*(1.03)^n
f(0) = 12*(1.03)^0
f(0) = 12*(1)
f(0) = 12
on day n = 0, aka the starting point, the height f(n) is 12 cm
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part c
answer: the average rate of change is approximately 0.43061036458991 (round however you need to). this represents the average growth rate from day n = 3 to day n = 10. so the plant grew roughly 0.43 cm per day during this timespan, assuming you round to 2 decimal places.
work shown:
compute f(3)
f(n) = 12*(1.03)^n
f(3) = 12*(1.03)^3
f(3) = 13.112724 < we'll use this later
compute f(10)
f(n) = 12*(1.03)^n
f(10) = 12*(1.03)^10
f(10) = 16.1269965521294 < we'll use this later
now use the formula below with a = 3 and b = 10
aroc = average rate of change
aroc = [ f(b) - f(a) ]/[ b - a ]
aroc = [ f(10) - f(3) ]/[ 10 - 3 ]
aroc = (16.1269965521294 - 13.112724)/(10 - 3)
aroc = 3.0142725521294/7
aroc = 0.43061036458991
round this however you need to
note: the plant grew approximately 3.01 cm over 7 days, so roughly 0.43 cm per day is the average growth rate (if you were to round to 2 decimal places).
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