Maths-

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Question

# A certain population of bacteria had an initial population of 1500. It then started growing at a rate of 2000 per day. Write the Application equation to model this situation. What are the slope and y-intercept? Explain.

Hint:

### 1. Growth in population of bacteria = Number of days of growth × rate of growth per day

2. Total population of bacteria = y = Initial population + Growth in population.

3. The slope of a line can be defined as the change in y coordinates of any 2 points on that line corresponding to the change in the x coordinates of those 2 points. This is generally referred to as the rise to run ratio of the given line i.e. how much did the y-coordinates rise vis-a-vis how long a distance was covered by the x-coordinates.

4. y-intercept for any line refers to the point at which the given line intersects with the Y-axis. At this point the x-coordinate is 0.

## The correct answer is: The given line can be written as- y = 2,000 x + 1,500. Its slope (i.e. change in population for a given change in time period) is 2,000. Its Y-intercept (i.e. population at 0 days or before starting growth) is 1,500.

### Step-by-step solution:-

Let the number of days for which the population of bacteria is observed to grow be x days and the total population of bacteria be y.

Now,

Growth in population of bacteria = Number of days of growth * rate of growth per day

∴ Growth in population of bacteria = x × 2,000 per day .......................................................................... (From given information)

∴ Growth in population of bacteria = 2,000 x ........................................................................................... (Equation i)

Now, we know that-

Total population of bacteria = y = Initial population + Growth in population.

∴ Total population of bacteria = y = 1,500 + 2,000 x .......................................................... (From given information & Equation i)

i.e. y = 2,000 x + 1,500 ................................................................................................................................. (Equation ii)

Comparing Equation ii with slope intercept form of a line i.e. y = mx + b, we get-

Slope = m = 2,000

& Y-intercept = b = 1,500

The slope of a line can be defined as the change in y coordinates of any 2 points on that line corresponding to the change in the x coordinates of those 2 points. This is generally referred to as the rise to run ratio of the given line i.e. how much did the y-coordinates rise vis-a-vis how long a distance was covered by the x-coordinates.

In the given situation, slope of the given equation refers to the movement in population of bacteria for every increase in the days that it is continues to grow.

since the slope = 2,000, we can say that for every day that the bacteria continues to grow, its population count increases by 2,000.

Also, y-intercept for any line refers to the point at which the given line intersects with the Y-axis.

At this point the x-coordinate is 0.

This means that when the value of x-coordinate is 0, the value of y-coordinate is 1,500.

For the given situation, this means that the population of bacteria, at the start i.e. befor any growth happened, will be 1,500.

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