Question

1. the table of values to the right shows the number of bacteria,y, in a sample during a science experiment after x minutes have elapsed.

a. find the number of bacteria remaining after 21 minutes have elapsed.

b. how many bacteria were in the sample to start the experiment?

Explanation:

Step1: Find the slope of the linear relationship

The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let \((x_1,y_1)=(8,90)\) and \((x_2,y_2)=(10,80)\).
\(m=\frac{80 - 90}{10 - 8}=\frac{-10}{2}=- 5\)

Step2: Find the equation of the line in point - slope form \(y - y_1=m(x - x_1)\)

Using \((x_1,y_1)=(8,90)\) and \(m=-5\), we have \(y-90=-5(x - 8)\)
Expand: \(y-90=-5x + 40\)
\(y=-5x+130\)

Part a

Step1: Substitute \(x = 21\) into the equation \(y=-5x + 130\)

\(y=-5\times21+130\)
\(y=-105 + 130\)
\(y = 25\)

Part b

Step1: Find the value of \(y\) when \(x = 0\) (start of the experiment)

Substitute \(x = 0\) into \(y=-5x+130\)
\(y=-5\times0+130\)
\(y = 130\)

Answer:

a. \(25\)
b. \(130\)