Question

1. a team of scientists began studying the population of white - tailed deer in a large wooded area in 2005. in 2007 (t = 2), there were 31 white - tailed deer in the wooded area. in 2020 (t = 15), there were 136 white - tailed deer. the number of white - tailed deer in the wooded area can be modeled by the function d given by d(t)=ab^t, where d(t) is the number of white - tailed deer during year t, and t is the number of years since 2005.

(a) (i) use the given data to write two equations that can be used to find the values for constants a and b in the expression for d(t).
(ii) find the values for a and b as a decimal approximation.
(b) (i) use the given data to find the average rate of change in the number of white - tailed deer, in deer per year, from t = 2 to t = 15 years. express your answer as a decimal approximation. show the computations that lead to your answer.
(ii) use the average rate of change found in (i) to estimate the number of white - tailed deer in the wooded area for t = 19 years. show the computations that lead to your answer.
(iii) consider the values that result from using the average rate of change found in (i) to estimate the number of white - tailed deer in the wooded area for times t = p years, where 2 < p < 15. are these estimates less than or greater than the number of white - tailed deer predicted by the model d at time t = p years. explain your reasoning. your explanation should include characteristics of the average rate of change and a reference to the graph of d.
(c) for which t - value, t = 8 years or t = 23 years, should the team of scientists have more confidence in when using the model d? give a reason for your answer in the context of the problem.
write your responses to this question only on the designated pages in the separate free response booklet.
write your solution to each part in the space provided for that part.
end of part a
if you finish before time is called,
you may check your work on part a only.
do not go on to part b until you are told to do so.

Explanation:

Part (A)(i)

Step1: Substitute \( t = 2 \)

Given \( D(t)=ab^t \), when \( t = 2 \) and \( D(2)=31 \), we get \( 31 = ab^{2} \).

Step2: Substitute \( t = 15 \)

When \( t = 15 \) and \( D(15)=136 \), we get \( 136 = ab^{15} \).

Step1: Solve for \( a \) from first equation

From \( 31 = ab^{2} \), we can express \( a=\frac{31}{b^{2}} \).

Step2: Substitute \( a \) into second equation

Substitute \( a=\frac{31}{b^{2}} \) into \( 136 = ab^{15} \), we have \( 136=\frac{31}{b^{2}}\times b^{15} \). Simplify the right - hand side: \( \frac{31}{b^{2}}\times b^{15}=31b^{15 - 2}=31b^{13} \). So, \( 136 = 31b^{13} \).

Step3: Solve for \( b \)

First, divide both sides by 31: \( b^{13}=\frac{136}{31}\approx4.3871 \). Then take the 13 - th root of both sides. Using a calculator, \( b=\sqrt[13]{\frac{136}{31}}\approx4.3871^{\frac{1}{13}}\approx1.12 \) (rounded to two decimal places).

Step4: Solve for \( a \)

Substitute \( b\approx1.12 \) into \( a = \frac{31}{b^{2}} \). \( b^{2}=(1.12)^{2}=1.2544 \), then \( a=\frac{31}{1.2544}\approx24.71 \) (rounded to two decimal places).

Step1: Recall the formula for average rate of change

The formula for the average rate of change of a function \( y = f(x) \) from \( x = x_1 \) to \( x = x_2 \) is \( \frac{f(x_2)-f(x_1)}{x_2 - x_1} \).

Step2: Substitute the values

Here, \( x_1 = 2 \), \( f(x_1)=31 \), \( x_2 = 15 \), \( f(x_2)=136 \). So the average rate of change \( r=\frac{136 - 31}{15 - 2}=\frac{105}{13}\approx8.08 \) (rounded to two decimal places).

Answer:

The two equations are \( 31 = ab^{2} \) and \( 136 = ab^{15} \).

Part (A)(ii)