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Answer:
(c)
Step-by-step explanation:
Given
See attachment for A and B
Required
Compare A and B
First, we get the initial population of A and B.
The initial population is at when [tex]t =0[/tex]
From the table of bacteria A, we have:
[tex]Initial = 100[/tex] when [tex]t = 0[/tex]
From the graph of bacteria B, we have:
[tex]Initial = 75[/tex] when [tex]t = 0[/tex]
Since the initial of bacteria B is less than that of bacteria A, then (a) is incorrect.
Next, calculate the slope of A and B i.e. the rate
Slope (m) is calculated as:
[tex]m = \frac{y_2 - y_1}{t_2 - t_1}[/tex]
Where
y = Number of bacteria
t = time
For bacteria A:
[tex](t_1,y_1) = (0,100)[/tex]
[tex](t_2,y_2) = (2,140)[/tex]
So, the slope is:
[tex]m_A = \frac{140 - 100}{2 - 0}[/tex]
[tex]m_A = \frac{40}{2}[/tex]
[tex]m_A = 20[/tex]
For bacteria B:
[tex](t_1,y_1) = (0,75)[/tex]
[tex](t_2,y_2) = (1,100)[/tex]
So, the slope is:
[tex]m_B = \frac{100- 75}{1 - 0}[/tex]
[tex]m_B = \frac{25}{1 }[/tex]
[tex]m_B = 25[/tex]
Since [tex]m_B > m_A[/tex], then the rate of bacteria B is greater than that of bacteria A.
Hence, (d) cannot be true
Next, we determine the equation of both bacteria
This is calculated using:
[tex]y = m(t - t_1) + y_1[/tex]
For bacteria A, we have:
[tex]y = m_A(t - t_1) + y_1[/tex]
Where:
[tex](t_1,y_1) = (0,100)[/tex]
[tex]m_A = 20[/tex]
So:
[tex]y = 20(t - 0) +100[/tex]
[tex]y = 20(t) +100[/tex]
[tex]y = 20t +100[/tex]
For bacteria B, we have:
[tex]y = m_B(t - t_1) + y_1[/tex]
Where:
[tex](t_1,y_1) = (0,75)[/tex]
[tex]m_B = 25[/tex]
So:
[tex]y = 25(t - 0) + 75[/tex]
[tex]y = 25(t) + 75[/tex]
[tex]y = 25t + 75[/tex]
At 3 hours, the population of bacteria A is:
[tex]y = 20t +100[/tex]
[tex]y = 20* 3 + 100[/tex]
[tex]y = 60 + 100[/tex]
[tex]y = 160[/tex]
At 3 hours, the population of bacteria B is:
[tex]y = 25t + 75[/tex]
[tex]y=25 * 3 + 75[/tex]
[tex]y=75 + 75[/tex]
[tex]y=150[/tex]
After 3 hours, bacteria B is 150 while A is 160.
This implies that (c) is correct because the population of B is less than that of A, at 3 hour
Lastly, to check if they will ever have equal population or not, we simply equate both equations.
So, we have:
[tex]y = y[/tex]
[tex]25t + 75 =20t + 100[/tex]
Collect like terms
[tex]25t - 20t = 100 - 75[/tex]
[tex]5t = 25[/tex]
Solve for t
[tex]t = 25/5[/tex]
[tex]t = 5[/tex]
They will have equal population at 5 hours.
Hence, b is incorrect
From the above computation, only (c) is correct
