Question

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

Answer 1

Answer:

7th Month

Step-by-step 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.