The population of a city increases by 4,000 people each year. In 2025, the population is projected to be 450,000 people. What is an equation that gives the city’s population p (in thousands of people) x years after 2010?
p = 4x + 50
p – 450 = 4(x – 15)
p – 15 = 4(x – 450)
p = 4x + 15
p = 4x + 15
Can you explain why?
Sure!
To find an equation that gives the city's population p (in thousands of people) x years after 2010, we need to consider the initial population in 2010 and how it changes over time.
We know that the population increases by 4,000 people each year. Since the population is given in thousands of people, we can say that the population increases by 4,000/1000 = 4 per year.
In 2010, the population is not mentioned, so we need to determine a constant term in the equation. We can calculate it by subtracting the population increase from the projected population in 2025.
450,000 - 4,000*15 = 450,000 - 60,000 = 390,000
So, in 2010, the population was 390,000 people (in thousands of people).
Now, let's write the equation using the slope-intercept form, which is y = mx + b, with y as the population, x as the number of years after 2010, m as the slope (rate of increase/ decrease), and b as the initial population.
Therefore, the equation becomes: p = 4x + 390 (where p represents the population and x represents the number of years after 2010).
To further simplify the equation, we can write it as p = 4x + 15 (since 390 = 4*15 + 15).
I hope this clarifies the reasoning behind the equation!
Can you write the answer in point slope form?
Certainly! The equation in point-slope form is:
p – 390 = 4(x – 0)
Alternatively, if you want to write the equation in point-slope form using 2010 as the starting point, it would be:
p – 390 = 4(x – 10)
What would it be using 2025 as the starting point?
If we use 2025 as the starting point, the equation in point-slope form would be:
p – 450 = 4(x – 15)
Could that be considered the answer of the question?
Apologies for the confusion. Yes, the equation p – 450 = 4(x – 15) can be considered as the answer to the question. It gives the city's population p (in thousands of people) x years after 2010, with 2025 as the starting point.
To determine the equation that gives the city's population, we need to consider the given information.
We know that the population increases by 4,000 people each year and that in 2025 (which is 15 years after 2010), the population is projected to be 450,000 people.
Since we want to express the population in thousands of people, we divide the population by 1,000.
Let's break down the problem step by step:
1. Let x be the number of years after 2010.
2. Each year, the population increases by 4,000 people. So, for each year, we add 4,000 to the initial population.
3. In 2010, the population is not given. Therefore, we need to find the initial population.
4. The population in 2025 is projected to be 450,000 people, which is 450,000/1,000 = 450 thousands of people.
5. Using the information above, we can write the equation for the population p (in thousands of people) x years after 2010.
Now, let's analyze each of the given equations:
a. p = 4x + 50
This equation seems incorrect because it adds 50 to the population for each year. However, the problem states that the population increases by 4,000 people each year, not by 50. So, this equation is not valid for this scenario.
b. p – 450 = 4(x – 15)
This equation is also incorrect because it assumes that the year 2010 is the reference point, but the problem states that the reference point is "x years after 2010." So, this equation is not valid either.
c. p – 15 = 4(x – 450)
This equation is also incorrect because it incorrectly assumes that the population decreases by 15 for each year. However, the problem states that the population increases by 4,000 people per year, not by 15.
d. p = 4x + 15
This equation is correct because it considers the correct reference point (2010) and correctly represents the population growth. It states that the population p (in thousands) is equal to 4 (the increase per year) multiplied by the number of years x, added to the initial population.
Thus, the correct equation that gives the city's population p (in thousands of people) x years after 2010 is p = 4x + 15.