Question

sheila is a wildlife biologist. at the beginning of each year, she tracks the wild turkey and white - tail deer population in the game reserve where she works.

• at the first year sheila counted 12 wild turkeys, and their number increases by approximately 40% each year.
• at the first year sheila counted 18 white - tail deer, and their number increases by 10 additional deer per year.

what is the first year in which sheila counts more turkeys than deer?

Explanation:

Step1: Define population functions

Let $n$ = number of years after Year 1.
Turkey population (exponential growth): $T(n) = 12(1+0.4)^n = 12(1.4)^n$
Deer population (linear growth): $D(n) = 18 + 10n$

Step2: Test Year 1 (n=0)

$T(0)=12(1.4)^0=12$, $D(0)=18+10(0)=18$
$12 < 18$ (turkeys < deer)

Step3: Test Year 2 (n=1)

$T(1)=12(1.4)^1=16.8$, $D(1)=18+10(1)=28$
$16.8 < 28$ (turkeys < deer)

Step4: Test Year 3 (n=2)

$T(2)=12(1.4)^2=12(1.96)=23.52$, $D(2)=18+10(2)=38$
$23.52 < 38$ (turkeys < deer)

Step5: Test Year 4 (n=3)

$T(3)=12(1.4)^3=12(2.744)=32.928$, $D(3)=18+10(3)=48$
$32.928 < 48$ (turkeys < deer)

Step6: Test Year 5 (n=4)

$T(4)=12(1.4)^4=12(3.8416)=46.0992$, $D(4)=18+10(4)=58$
$46.0992 < 58$ (turkeys < deer)

Step7: Test Year 6 (n=5)

$T(5)=12(1.4)^5=12(5.37824)=64.53888$, $D(5)=18+10(5)=68$
$64.53888 < 68$ (turkeys < deer)

Step8: Test Year 7 (n=6)

$T(6)=12(1.4)^6=12(7.529536)=90.354432$, $D(6)=18+10(6)=78$
$90.354432 > 78$ (turkeys > deer)

Answer:

Year 7