IDNLearn.com: Your trusted platform for finding reliable answers. Our experts provide timely and precise responses to help you understand and solve any issue you face.
Sagot :
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]
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]
We appreciate your contributions to this forum. Don't forget to check back for the latest answers. Keep asking, answering, and sharing useful information. IDNLearn.com is your source for precise answers. Thank you for visiting, and we look forward to helping you again soon.