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

Part A:

Let n represent the number of years.

Neighborhood A: f(n) = 30 * 1.2^n

Neighborhood B: g(n) = 45 + 3n

Part B:

Neighborhood A has 81 homes after 5 years. Neighborhood B has 60 homes after 5 years.

Part C:

We can set the two functions equal to each other and solve for n:

30 * 1.2^n = 45 + 3n

1.2^n = (45 + 3n) / 30

Taking the logarithm of both sides, we get:

n * log(1.2) = log((45 + 3n) / 30)

n * 0.18232 = log((45 + 3n) / 30)

n = (log((45 + 3n) / 30)) / 0.18232

Solving this equation numerically, we find that the number of homes in Neighborhood A and Neighborhood B will be the same after approximately 11.7 years.

Answer 2

Explanation:

Part A:

Let A(t) be the number of homes in Neighborhood A after t years, and let B(t) be the number of homes in Neighborhood B after t years.

For Neighborhood A, the number of homes increases by 20% each year. This can be represented by the following formula:

A(t) = 30(1 + 0.2)^t

For Neighborhood B, 3 new homes are built each year. This can be represented by the following formula:

B(t) = 45 + 3t

Part B:

To find the number of homes in Neighborhood A and Neighborhood B after 5 years, we can substitute t = 5 into the respective formulas:

A(5) = 30(1 + 0.2)^5 = 83.66 homes (rounded to two decimal places)

B(5) = 45 + 3(5) = 60 homes

Therefore, Neighborhood A has 83.66 homes after 5 years, and Neighborhood B has 60 homes after 5 years.

Part C:

We need to find the value of t such that A(t) = B(t).

Substituting the expressions for A(t) and B(t), we get:

30(1 + 0.2)^t = 45 + 3t

Dividing both sides by 3, we get:

10(1 + 0.2)^t = 15 + t

Taking the natural logarithm of both sides, we get:

ln(10(1 + 0.2)^t) = ln(15 + t)

Using the properties of logarithms, we can simplify this to:

t = ln(10/3) / ln(1.2) + ln(5)

Using a calculator, we get:

t ≈ 7.63

Therefore, after approximately 7.63 years, the number of homes in Neighborhood A and Neighborhood B will be the same.