Option a: The number of bacteria at time x is 0.
Option b: An exponential function that represents the population is [tex]y=200(1.5)^x[/tex]
Option c: The population after 10 minutes is 11534(app)
Explanation:
It is given that the coordinates of the graph are (0,200), (1,300) and (2, 450)
Option a: To determine the number of bacteria x when y = 200
From the graph, we can see that the line meets y = 200 when x = 0
Thus, the coordinates are (0,200)
Hence, the number of bacteria at time x is 0 when y = 200.
Option b: Now, we shall determine the exponential function of the population.
The general formula for exponential function is [tex]y=a \cdot b^{x}[/tex]
Where a is the starting point and [tex]a=200[/tex]
b is the common difference.
To determine the common difference, let us divide,
[tex]\frac{300}{200} =1.5[/tex]
Also, [tex]\frac{450}{300} =1.5[/tex]
Hence, the common difference is [tex]b=1.5[/tex]
Thus, substituting the values [tex]a=200[/tex] and [tex]b=1.5[/tex] in the formula [tex]y=a \cdot b^{x}[/tex],
we have, [tex]y=200(1.5)^x[/tex]
Hence, An exponential function that represents the population is [tex]y=200(1.5)^x[/tex]
Option c: To determine the population after 10 minutes, let us substitute [tex]x=10[/tex] in [tex]y=200(1.5)^x[/tex], since the x represents the population of the bacteria in minutes.
Thus, we have,
[tex]\begin{aligned}y &=200(1.5)^{x} \\&=200(1.5)^{10} \\&=200(57.67) \\&=11534\end{aligned}[/tex]
Hence, the population after 10 minutes is 11534(app)
