Question

1. a group of biologists is surveying the bat population in a cave. the equation ( b(t)=60 cdot 2.5^{t} ) gives the total number of bats, ( b ), ( t ) years since 2020.

a. explain what the numbers 60 and 2.5 represent in the context of this situation. (2 pts)
( 60=)____
( 2.5=)____
b. predict the number of bats, ( b ), in the cave after 5 years. show your work. (2 points)

Explanation:

Step1: Analyze the function type

The function \( b(t)=60\times2.5^{t}\) is in the form of \( y = a\times b^{x}\) (where \(a = 60\), \(b=2.5\)), which is an exponential function.

Step2: Substitute \(t = 5\) into the function

When \(t = 5\), we have \(b(5)=60\times2.5^{5}\).
First, calculate \(2.5^{5}=2.5\times2.5\times2.5\times2.5\times2.5=(2.5\times2.5)\times(2.5\times2.5)\times2.5 = 6.25\times6.25\times2.5=39.0625\times2.5 = 97.65625\).
Then, \(b(5)=60\times97.65625\).
\(60\times97.65625=(60\times97)+(60\times0.65625)=5820 + 39.375=5859.375\).

Answer:

a. The number \(60\) represents the initial number of bats (the population of bats at \(t = 0\) i.e., in the year 2020). The number \(2.5\) represents the growth factor. For each year \(t\), the bat - population is multiplied by \(2.5\).
b. \(b(5)=60\times2.5^{5}=60\times97.65625 = 5859.375\)