Question

The population of frogs in a park is decreasing exponentially at a rate of 6% each year. In the year 2007, the population was estimated to be 5500. a. Use the exponential decay formula to write an equation to model the situation. b. Use your equation from Part a to predict the frog population in the year 2018. c. Sketch the graph of the equation you wrote in Part a. Find its asymptote, domain, and range.

Answer 1

a. Since, the exponential decay formula is,

`y=a(1-r)^x`

Where, a is the initial value,

r is the decay rate per period,

x is the number of periods,

Suppose, the population is estimated from the year 2007,

Given,

The population in 2007 is, a = 5500,

Annual rate of decay, r = 6% = 0.06,

Hence, the population after x years,

`y=5500(1-0.06)^x`

`\implies y=5500(0.94)^x`

Which is the required equation.

b. for 2018, x = 11 ( number of years since 2007 ),

Thus, the population in 2018 would be,

`y=5500(0.94)^(11)`

`=2784.64013987`

`\approx 2785`

c. The graph of
`y=5500(0.94)^(x)` shown below,

Having no vertical asymptote,

And, horizontal asymptote is y=0

Domain = All possible value of x = x ,

Range = All possible value of y = 0 ≤ y ≤ 5500

Answer 2

y = a(1-b)^x
y= final amount
a=original amount
x=time
b= percent change(it would be + if it said increasing) note:change to decimal form.
y=5500 x(1-6%)^(2018-2007)
y= 5500(.94^11)