Let's solve the given problem step-by-step:
Step 1: Understand the given data and the question.
- The initial enrollment capacity is 120 students.
- The school plans to triple the enrollment capacity every year.
- The target enrollment capacity is 3,240 students.
- We need to determine how many years, [tex]\( t \)[/tex], it will take to achieve the target.
Step 2: Formulate the equation which represents this growth.
The school triples its enrollment capacity every year. This means we have exponential growth where each year the previous enrollment number is multiplied by 3. The mathematical representation for this continuous growth is:
[tex]\[ enrollment\_capacity = initial\_enrollment \times (3)^t \][/tex]
Given:
[tex]\[ enrollment\_capacity = 3,240 \][/tex]
[tex]\[ initial\_enrollment = 120 \][/tex]
Thus, the equation becomes:
[tex]\[ 120 \times (3)^t = 3,240 \][/tex]
Step 3: Solve for [tex]\( t \)[/tex].
- To isolate [tex]\( t \)[/tex], we take the ratio of the target enrollment to the initial enrollment:
[tex]\[ \frac{3,240}{120} = (3)^t \][/tex]
- Simplify the fraction:
[tex]\[ 27 = (3)^t \][/tex]
- Recognize that [tex]\( 27 = 3^3 \)[/tex]:
[tex]\[ (3)^3 = (3)^t \][/tex]
- Therefore,
[tex]\[ t = 3 \][/tex]
Answer:
After 3 years, the graduate school will be able to achieve its target enrollment capacity of 3,240 students. Therefore, the correct answer is:
[tex]\[ 120(3)^t = 3,240 ; t = 3 \][/tex]
Matched to the given options, the correct choice is:
C. [tex]\( 120(3)^t = 3,240 ; t = 3 \)[/tex]
