Question

What is the first month in which Maria's amount of land exceeds Lucia's amount of land?

Answer 1

7th Month

Explanation:

The number of hectares of Lucia's land grows arithmetically by 5 hectares each month.

Whereas, the number of hectares of Maria's land grows geometrically with a common ratio of 1.4 every month.

Writing this as sequence expressions:

For Lucia,

First term, a=11 hectares

Common difference, d=5

Therefore:

`T(n)=a+(n-1)d`

`11+5(n-1)=11+5n-5\\=6+5n`

For Maria,

First term, a=6 hectares

Commons ratio=1.4

`T(n)=ar^(n-1)\\=6*1.4^(n-1)`

Next, we compute and compare the results for values of n of these two equations:

`Lucia, T(n)=6+5n`

`Maria,T(n)=6*1.4^(n-1)`

`Lucia, T(1)=6+5=11`

`Maria,T(1)=6*1.4^(1-1)=6`

`Lucia, T(2)=6+5(2)=16`

`Maria,T(2)=6*1.4^(2-1)=8.4`

`Lucia, T(3)=6+5(3)=21`

`Maria,T(3)=6*1.4^(3-1)=11.76`

`Lucia, T(4)=6+5(4)=26`

`Maria,T(4)=6*1.4^(4-1)=16.56`

`Lucia, T(5)=6+5(5)=31`

`Maria,T(5)=6*1.4^(5-1)=23.05`

`Lucia, T(6)=6+5(6)=36`

`Maria,T(6)=6*1.4^(6-1)=32.27`

`Lucia, T(7)=6+5(7)=41`

`Maria,T(7)=6*1.4^(7-1)=45.18`

Therefore, in the seventh month, Maria's land will exceed that of Lucia.