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

(c)

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 = (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 = (140 - 100)/(2 - 0)`

`m_A = (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 = (100- 75)/(1 - 0)`

`m_B = (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