Question

a fish population, p, can be represented by the equation ( p = 800 cdot (\frac{1}{2})^t ) where t is time in years since the beginning of 2015. a. what was the fish population when ( t = -3 )? select the correct choice. a 2,400 b 6,400 c 800 d 100 b. what does this mean in this situation? the number above represents the population of fish in 2018. the number above represents the population of fish in 2012

Explanation:

Step1: Substitute \( t = - 3 \) into the equation

Given \( p = 800\cdot(\frac{1}{2})^{t} \), when \( t=-3 \), we have \( p = 800\cdot(\frac{1}{2})^{-3} \).
Using the rule \( a^{-n}=\frac{1}{a^{n}} \), then \( (\frac{1}{2})^{-3}=2^{3} \).

Step2: Calculate the value

Since \( 2^{3}=8 \), then \( p = 800\times8=6400 \).

Answer:

A. \( 2400 \) (Incorrect)
B. \( 6400 \) (Correct)
C. \( 800 \) (Incorrect)
D. \( 100 \) (Incorrect)

For part b:
Since \( t \) is time in years since the beginning of 2015, when \( t=-3 \), it means \( 2015-3 = 2012 \). So the number above represents the population of fish in 2012.