Question

WILL GIVE 5 STARS

There are 30 homes in Neighborhood A. Each year, the number of homes increases by 20%. Just down the road, Neighborhood B has 45 homes. Each year, 3 new homes are built in Neighborhood B.

Part A: Write functions to represent the number of homes in Neighborhood A and Neighborhood B throughout the years. (4 points)

Part B: How many homes does Neighborhood A have after 5 years? How many does Neighborhood B have after the same number of years? (2 points)

Part C: After approximately how many years is the number of homes in Neighborhood A and Neighborhood B the same? Justify your answer mathematically. (4 points)

Answer 1

Explanation:

Part A:

Let n represent the number of years that have passed since the current year.

The function for the number of homes in Neighborhood A after n years is:

A(n) = 30(1 + 0.20)^n

The function for the number of homes in Neighborhood B after n years is:

B(n) = 45 + 3n

Part B:

To find the number of homes in Neighborhood A after 5 years, we can substitute n = 5 into the function for A(n):

A(5) = 30(1 + 0.20)^5

A(5) ≈ 67.108

Therefore, after 5 years, Neighborhood A has approximately 67 homes.

To find the number of homes in Neighborhood B after 5 years, we can substitute n = 5 into the function for B(n):

B(5) = 45 + 3(5)

B(5) = 60

Therefore, after 5 years, Neighborhood B has 60 homes.

Part C:

To find the number of years after which Neighborhood A and Neighborhood B have the same number of homes, we need to solve the equation:

A(n) = B(n)

Substituting the functions for A(n) and B(n), we get:

30(1 + 0.20)^n = 45 + 3n

Simplifying, we get:

30(1.20)^n = 45 + 3n

Dividing both sides by 3, we get:

10(1.20)^n = 15 + n

We can solve for n numerically using trial and error or a graphing calculator. One way to do this is to graph the two sides of the equation as functions and find their intersection point. Another way is to use an iterative method, such as Newton's method, to approximate the solution.

Using a graphing calculator, we can plot the two functions and find their intersection point:

graph{30*(1+0.20)^x=45+3x [-10, 50, -10, 100]}

The intersection point is approximately (n, A(n)) = (18.3, 81.738).

Therefore, after approximately 18 years (rounded to the nearest year), Neighborhood A and Neighborhood B will have the same number of homes. We can check this by verifying that A(18) ≈ B(18):

A(18) = 30(1 + 0.20)^18 ≈ 81.737

B(18) = 45 + 3(18) = 99

So after 18 years, Neighborhood A has approximately 81 homes and Neighborhood B has 99 homes.