Question

A. How long does Milmo City have a smaller population compared to Ridge City?B. According to the linear equations, does Milmo City ever have a smaller population after the populations become equal? Explain your reasoning why.C. When is the population of Ridge City double the population of Milmo City?D. Explain why Milmo City has a smaller population at the start but will have a larger population after nine years compared to Ridge City?

Answer 1

Given the equations:

`\begin{gathered} P_M=(5)/(3)t+2 \\ \\ P_R=(2)/(3)t+6 \end{gathered}`

The equation Pm models the population for Milmo city.

The equation PR m0dels the population for Ridge city.

Let's graph the two populations for the first 10 years.

Let's graph the equations using two points each.

For the first equation:

When t = 3:

`\begin{gathered} P_M=(5)/(3)*3+2 \\ \\ P_M=5+2=7 \end{gathered}`

When t = 6:

`\begin{gathered} P_M=(5)/(3)*6+2 \\ \\ P_M=10+2 \\ \\ P_M=12 \end{gathered}`

For the first equation, we have the two points:

(3, 7) and (6, 12).

To graph the line, plot the points and connect the points with a straight line.

For equation 2:

When t = 3:

`\begin{gathered} P_R=(2)/(3)*3+6 \\ \\ P_R=2+6 \\ \\ P_R=8 \end{gathered}`

When t = 6:

`\begin{gathered} P_R=(2)/(3)*6+6 \\ \\ P_R=4+6 \\ \\ P_R=10 \end{gathered}`

We have the two points:

(3, 8) and (6, 10)

Plot the points and connect the points using a straight line.

We have the graph below:

The red line represents the population for MIlmo City.

The blue line represents the population for Ridge City.

Part A.

From the graph, we can see that both lines meet at the point (4, 8.667).

This means that Milmo City had a smaller population for 4 years.

The population became equal at 4 years.

This is because the red line (MIlmo City) is below the blue line(Ridge) till both lines intersect at t = 4.

After the intersection, the red line now goes above the blue line.

Part B.

After the populations became equal, Milmo goes above the population of Ridge City.

Therefore, Milmo City will never have a smaller population after the populations become equal.

Part C.

Let's determine when the population of Ridge CIty double the population of Milmo City.

We have:

`\begin{gathered} P_R=2P_M \\ \\ (2)/(3)t+6=2((5)/(3)t+2) \\ \\ (2)/(3)t+6=(10)/(3)t+2 \end{gathered}`

Let's solve for t:

`\begin{gathered} (10)/(3)t-(2)/(3)t=6-4 \\ \\ (10-2)/(3)t=2 \\ \\ (8)/(3)t=2 \\ \\ t=2*(3)/(8) \\ \\ t=(6)/(8) \\ \\ t=0.75 \end{gathered}`

Therefore, the population of Ridge City is double the population of Milmo City when the time is 0.75 years.

• Part D:

The population for Milmo City has a smaller population at the start because the y-intercept (2) is lower than the y-intercept for Ridge City (6), but on the long run the population of Milmo city will be greater than the population of Ridge City because the slope(rate of change) for Milmo City (5/3) is greater than the rate of change for Ridge City.

That is why after 9 years, the population of Milmo City is greater than the population of Ridge City.

ANSWER:

• (A). Milmo City had a smaller population for 4 years.

• (B). Milmo City will never have a smaller population after the populations become equal.

• (C),. T,he population of Ridge City is double the population of Milmo City when the time is 0.75 years.

• (D). The population for Milmo City has a smaller population at the start because the y-intercept (2) is lower than the y-intercept for Ridge City (6), but on the long run the population of Milmo city will be greater than the population of Ridge City because the slope(rate of change) for Milmo City (5/3) is greater than the rate of change for Ridge City.