Question

in 2003, 116 deer were introduced into a wildlife refuge area from which the population had previously been hunted to elimination. by 2006, the population had grown to 200 deer. if this population grows exponentially, find the values of ( a ) and ( b ) in the formula for the function ( f(x) = a cdot b^x )( a = square ) and ( b = square )round ( a ) to the nearest deer, and round ( b ) to one decimal.

Explanation:

Step1: Define x as years since 2003

Let $x = 0$ represent 2003, $x=3$ represent 2006.

Step2: Find initial value a

At $x=0$, $f(0)=a\cdot b^0=a=116$.

Step3: Substitute 2006 values to find b

Substitute $x=3$, $f(3)=200$, $a=116$:
$$200 = 116 \cdot b^3$$
Solve for $b^3$:
$$b^3 = \frac{200}{116} \approx 1.7241$$
Take cube root:
$$b = \sqrt[3]{1.7241} \approx 1.20$$

Step4: Round values as required

$a$ rounds to 116, $b$ rounds to 1.2.

Answer:

$a = 116$ and $b = 1.2$