To solve the given problem, let's start with the equation representing the membership of the student group:
[tex]\[ y = 14x + 10 \][/tex]
where [tex]\( y \)[/tex] represents the number of members and [tex]\( x \)[/tex] represents the number of years since the group was formed.
We are tasked with finding the number of years it will take for the group to have 290 members. Given [tex]\( y = 290 \)[/tex], we can set up the equation:
[tex]\[ 290 = 14x + 10 \][/tex]
First, let's isolate [tex]\( x \)[/tex]. Subtract 10 from both sides of the equation:
[tex]\[ 290 - 10 = 14x \][/tex]
[tex]\[ 280 = 14x \][/tex]
Next, divide both sides by 14 to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{280}{14} \][/tex]
[tex]\[ x = 20 \][/tex]
So, it will take 20 years for the group to have 290 members.
Next, if the group was formed in the year 2000, we need to determine the actual year when the group will have 290 members. Since it takes 20 years from the formation, we add these 20 years to the year 2000:
[tex]\[ \text{Year} = 2000 + 20 \][/tex]
[tex]\[ \text{Year} = 2020 \][/tex]
Thus, the group will have 290 members in the year 2020.
Summarizing the answers:
- The number of years it will take for the group to have 290 members is 20 years.
- The year in which the group will have 290 members is 2020.
