Question

The populations of two cultures of bacteria, A and B, after x hours are shown below. Which statement is a correct comparison of bacteria A and bacteria B?

Answer 1

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 $t =0$

From the table of bacteria A, we have:

$Initial = 100$ when $t = 0$

From the graph of bacteria B, we have:

$Initial = 75$ when $t = 0$

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:

$m = \frac{y_2 - y_1}{t_2 - t_1}$

Where

y = Number of bacteria

t = time

For bacteria A:

$(t_1,y_1) = (0,100)$

$(t_2,y_2) = (2,140)$

So, the slope is:

$m_A = \frac{140 - 100}{2 - 0}$

$m_A = \frac{40}{2}$

$m_A = 20$

For bacteria B:

$(t_1,y_1) = (0,75)$

$(t_2,y_2) = (1,100)$

So, the slope is:

$m_B = \frac{100- 75}{1 - 0}$

$m_B = \frac{25}{1 }$

$m_B = 25$

Since $m_B > m_A$, 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:

$y = m(t - t_1) + y_1$

For bacteria A, we have:

$y = m_A(t - t_1) + y_1$

Where:

$(t_1,y_1) = (0,100)$

$m_A = 20$

So:

$y = 20(t - 0) +100$

$y = 20(t) +100$

$y = 20t +100$

For bacteria B, we have:

$y = m_B(t - t_1) + y_1$

Where:

$(t_1,y_1) = (0,75)$

$m_B = 25$

So:

$y = 25(t - 0) + 75$

$y = 25(t) + 75$

$y = 25t + 75$

At 3 hours, the population of bacteria A is:

$y = 20t +100$

$y = 20* 3 + 100$

$y = 60 + 100$

$y = 160$

At 3 hours, the population of bacteria B is:

$y = 25t + 75$

$y=25 * 3 + 75$

$y=75 + 75$

$y=150$

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:

$y = y$

$25t + 75 =20t + 100$

Collect like terms

$25t - 20t = 100 - 75$

$5t = 25$

Solve for t

$t = 25/5$

$t = 5$

They will have equal population at 5 hours.

Hence, b is incorrect

From the above computation, only (c) is correct