Question

select all the correct answers.
every year, a town surveys the size of its wetland ecosystems. this equation estimates the total wetland area, in acres, over time, ( t ), in years
( f(t)=740(0.95)^{t} )
which statements are supported by the function?
when ( t = 10 ), the town is predicted to have around 443 acres of wetlands.
at one time, the town had 740 acres of wetlands.
the wetland area in the town decreases by ( 5 % ) per year.
the wetland area in the town increases by ( 95 % ) per year.
when ( t = 10 ), the town is predicted to have around 492 acres of wetlands.

Explanation:

Step1: Analyze the function form

The function \( f(t)=740(0.95)^{t} \) is in the form of an exponential decay function \( y = a(1 - r)^{t} \), where \( a = 740 \) (initial value) and \( r=1 - 0.95=0.05 = 5\%\) (rate of decrease).

Step2: Check the initial - value statement

When \( t = 0 \), \( f(0)=740(0.95)^{0}=740\times1 = 740 \). So, at one time (when \( t = 0 \)), the town had 740 acres of wetlands.

Step3: Check the rate of change statement

Since the base of the exponential function \( 0.95=1 - 0.05 \), the wetland area decreases by \( 5\%\) per year (\(r = 5\%\)).

Step4: Calculate \( f(t) \) when \( t = 10 \)

Substitute \( t = 10 \) into \( f(t)=740(0.95)^{t} \). Then \( f(10)=740\times(0.95)^{10}\).
Using a calculator, \( (0.95)^{10}\approx0.5987 \), and \( f(10)=740\times0.5987\approx443 \).

Answer:

• When \( t = 10 \), the town is predicted to have around 443 acres of wetlands.
• At one time, the town had 740 acres of wetlands.
• The wetland area in the town decreases by 5% per year.