Between 2000 and 2020, the population of Mathville could be modeled by the function m(t)=100(3(sq. root)t), where m(t) is the number of people in Mathville, and t is the number of years since 2000. Between those same years, the population of Calcfield could be modeled by the function c(t)=18t

Question

A: Graph the function on graph paper or a neatly made coordinate grid by hand. be sure to consider an appropriate domain for the functions as you make your graph.
B: Approximately where do the functions intersect? What does this point of intersection represent?
C: Write and solve an equation to algebraically confirm where the two functions intersect. Show your work.
D: Write 2-3 complete sentences comparing the relative populations of the cities over time.

Answer 1

To work through the problem systematically, we’ll break down each part.

A: Graph the Functions

You need to graph:

1. Mathville Population: \( m(t) = 100(3\sqrt{t}) \)
2. Calcfield Population: \( c(t) = 18t \)

Choosing a domain:

Since the functions model the population between the years 2000 to 2020, we can set our domain for \( t \) from 0 to 20 (representing the years 2000 to 2020).

Points to Evaluate:

• For Mathville (\( m(t) = 100(3\sqrt{t}) \)):

  • \( t = 0 \): \( m(0) = 100(3\sqrt{0}) = 0 \)
  • \( t = 1 \): \( m(1) = 100(3\sqrt{1}) = 300 \)
  • \( t = 4 \): \( m(4) = 100(3\sqrt{4}) = 600 \)
  • \( t = 9 \): \( m(9) = 100(3\sqrt{9}) = 900 \)
  • \( t = 16 \): \( m(16) = 100(3\sqrt{16}) = 1200 \)
  • \( t = 20 \): \( m(20) = 100(3\sqrt{20}) \approx 1344.49 \)

• For Calcfield (\( c(t) = 18t \)):

  • \( t = 0 \): \( c(0) = 18(0) = 0 \)
  • \( t = 1 \): \( c(1) = 18(1) = 18 \)
  • \( t = 4 \): \( c(4) = 18(4) = 72 \)
  • \( t = 9 \): \( c(9) = 18(9) = 162 \)
  • \( t = 16 \): \( c(16) = 18(16) = 288 \)
  • \( t = 20 \): \( c(20) = 18(20) = 360 \)

You will need to plot these points on a graph covering the domain from \( t = 0 \) to \( t = 20 \) for both functions. On the y-axis, use appropriate scale to accommodate the largest value, which is slightly above 1344 for Mathville.

B: Approximate Intersection

To approximate where the functions intersect, you can observe the figures you've constructed on the graph. Visually inspect where the two curves cross.

From the plotted points:

• At \( t = 16 \):

  • \( m(16) = 1200 \)
  • \( c(16) = 288 \)

• Between \( t = 16 \) and \( t = 20 \), the values of \( m(t) \) increase rapidly whereas \( c(t) \) increases linearly.

The intersection occurs somewhere between \( t = 16 \) and \( t = 20 \).

This intersection represents the year when Mathville and Calcfield have the same population.

C: Algebraic Solution to Find Intersection

To find the intersection point algebraically, set the two equations equal to each other:

\[ 100(3\sqrt{t}) = 18t \]

1. Simplify the equation: \[ 300\sqrt{t} = 18t \]

2. Rearranging gives: \[ 300\sqrt{t} - 18t = 0 \] \[ \sqrt{t}(300 - 18\sqrt{t}) = 0 \]

So, we have two solutions:

• \( \sqrt{t} = 0 \) which means \( t = 0\) (the starting point)
• \( 300 - 18\sqrt{t} = 0 \)

3. Solving for \( \sqrt{t} \): \[ 18\sqrt{t} = 300 \] \[ \sqrt{t} = \frac{300}{18} = 16.67 \] \[ t \approx (16.67)^2 \approx 278.89 \] (no year value)

This value indicates an algebraic mistake, let’s compute for \( \sqrt{t} \): Assuming \( \sqrt{t} = x \), gives \( t = x^2 \) \[ 300 - 18x = 0 \ x \approx 16.67 \ t = (16.67)^2 \text{ (invalid since we need to limit)} \]

We continue with assumptions until reaching specific \( t \): Plugging estimates until we reach equivalent where Uses numerical methods until \( m(t) = c(t) \).

D: Population Comparison

As observed, from the year 2000 to 2020, Mathville's population growth, modeled by a square root function, begins slowly but accelerates significantly over time. In contrast, Calcfield's linear growth reflects a steady increase. Eventually, the growth rates suggest that Mathville's population surpasses Calcfield's population, indicating a more dynamic growth within its community, suggesting possible urban development policies or increased resource allocation boosting the living conditions.

Note: Ensure graphs are properly interpreted once plotted for accurate comparison and intersection solutions.