you are planting roses
you started at 12 noon
at 3pm, you have 80 roses left
at 6pm, you have 20 roses left
if you planted roses at a constant rate, how many roses did you start with?
so we can find the slope of this linear function as (20roses-80roses)/(6pm-3pm)=60roses/3hrs=-20roses/hr
your quantity goes down by 20 roses per hour (you plant 20 per hour)
r=-20t+?
find ?, the number of roses when t=0
at 3pm yo have 80 roses
80=-20(3)+?
80=-60+?
140=?
you had 140 roses intiailly
if you wanted an exponenntial
the town polpulationn grows at an exponential rate
the town was founded in 2000
in 2010, the town's population was 1000
in 2015, the town's populationw as 2000
find intial polpulation
exponential can be modeled by [tex]y=ab^t[/tex] where y=population, t=time and b and a are constants (a=initial population)
we are given that after 10 years (2010), y=102400
after 15 years, (2015), y=3276800
so
[tex]102400=ab^{10}[/tex] and [tex]3276800=ab^{15}[/tex]
divide the 2nd by the first equation, remembering our exponential properties like [tex]\frac{x^a}{x^b}=x^{a-b}[/tex]
[tex]32=b^{5}[/tex]
take 5th root of both sides
[tex]b=2[/tex]
subsitute back
[tex]102400=ab^{10}[/tex]
[tex]102400=a(2)^{10}[/tex]
[tex]102400=1024a[/tex]
[tex]100=a[/tex]
the intial population is 100 people
