Question

(05.03 HC) 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.

Answer 1

Explanation:

From the information given:

Neighbourhood A = 30 homes and the number increases by 20% each year

Neighbourhood B = 45 homes, and each year 3 new homes are built.

A.

The function representing the numbers of homes in Neighbourhood A and B are as follows:

For neighbourhood A: f(x) =
`\mathbf{30 * (1.2)^x}`

For neighbourhood B: f(x) = 45 + 3x

B.

After five years;

Neighbourhood A has =
`\mathbf{30 * (1.2)^x}`

Neighbourhood A =
`\mathbf{30 * (1.2)^5}`

Neighbourhood A = 74.65 homes

Neighbourhood B: = 45 + 3(5)

Neighbourhood B: = 45 + 15

Neighbourhood B: = 60 homes

C.

To determine how many years the number of homes are the same for neighbourhood A and B, we need to equate both together.

i.e.

`\mathbf{30 * (1.2)^x= 45 + 3x}`

`\mathbf{ (1.2)^x= (45 + 3x)/(30)}`

`\mathbf{ (0.4)^x= (15 + x)/(10)}`

x = 3.3

Thus, after 3.3 years, the number of homes will be the same.