Let's break down the problem step by step to find the correct equation and the value of [tex]\( t \)[/tex].
1. Initial Enrollment Capacity: The current enrollment capacity is 120 graduate students.
2. Target Enrollment Capacity: The goal is to increase this capacity to 3,240 students.
3. Annual Growth Factor: The school plans to triple the enrollment every year. This means the enrollment capacity each year is multiplied by 3.
We are asked to find how many years, [tex]\( t \)[/tex], it will take to reach the target capacity of 3,240 students. We use the exponential growth formula:
[tex]\[
\text{Future Value} = \text{Present Value} \times (\text{Growth Rate})^t
\][/tex]
In this case:
[tex]\[
3240 = 120 \times 3^t
\][/tex]
Our task is to solve for [tex]\( t \)[/tex].
From pre-calculated data:
[tex]\[
120 \times 3^t = 3240
\][/tex]
Dividing both sides by 120:
[tex]\[
3^t = \frac{3240}{120}
\][/tex]
[tex]\[
3^t = 27
\][/tex]
We know that:
[tex]\[
27 = 3^3
\][/tex]
Thus:
[tex]\[
3^t = 3^3
\][/tex]
Since the bases are the same, the exponents must be equal:
[tex]\[
t = 3
\][/tex]
Therefore, it will take 3 years for the graduate school to achieve its target enrollment capacity of 3,240 students.
The correct equation representing this situation is:
[tex]\[
120 \times 3^t = 3240
\][/tex]
And after solving, we get:
[tex]\[
t = 3
\][/tex]
Now let's compare this to the provided options:
A. [tex]\((120-3)^t=3,240: t=2\)[/tex] - This equation is incorrect because it does not follow the exponential growth model specified in the problem.
B. [tex]\(120+(3)^2=3,240-1\)[/tex] - This equation is incorrect as it adds a constant growth factor rather than multiplying the enrollment capacity annually.
C. [tex]\(120(3)^t=3,240 ; t \leq 3\)[/tex] - This equation correctly represents an exponential growth model matching the problem, and with [tex]\( t = 3 \)[/tex], it is correct.
D. [tex]\(120(1.3)^4=3,240 ; t=27\)[/tex] - This equation is incorrect as it uses a different growth rate of 1.3 instead of 3, and the resulting exponent [tex]\( t \)[/tex] is not accurate.
Thus, the correct answer is:
C. [tex]\(120(3)^t=3,240 ; t \leq 3\)[/tex]
