Problem 112 - Calculus Volume 1

Question

ITI The admissions office at a public university estimates that 65\(\%\) of the students offered admission to the class of 2019 will actually enroll. a. Find the linear function \(y=N(x),\) where \(N\) is the number of students that actually enroll and \(x\) is the number of all students offered admission to the class of 2019 . b. If the university wants the 2019 freshman class size to be 1350 , determine how many students should be admitted.

Step-by-Step Solution

Answer

1. Linear function: \(y = 0.65x\). 2. Admit approximately 2077 students.

Step 1: Define the Problem

We are asked to find a linear function that represents the relationship between the number of students offered admission \(x\) and the number of students \(y\) who actually enroll. We know that 65\% of the students offered admission will enroll.

Step 2: Determine the Linear Function

To find the function \(y = N(x)\), note that \(y\) is 65\% of \(x\). Therefore, the linear function can be written as \(y = 0.65x\).

Step 3: Solve for Required Admissions

To find how many students need to be admitted to achieve a freshman class size of 1350, we set \(y\) to 1350 in the function \(y = 0.65x\). Then solve for \(x\):\[1350 = 0.65x\]Divide both sides by 0.65:\[x = \frac{1350}{0.65} \approx 2077\]

Step 4: Conclusion

Therefore, to achieve a freshman class size of 1350 students, the university should offer admission to approximately 2077 students.

Key Concepts

Understanding the Admissions Formula

The admissions formula is essentially a linear function that calculates how many students will actually enroll from those offered admission. In our specific scenario, the relationship between the number of students offered admission (denoted as \( x \)) and those who enroll (denoted as \( y \)) is linear. This is because we have a constant enrollment rate of 65\%.
This relationship can be represented as the linear equation \( y = 0.65x \). Here, \( 0.65 \) is the slope or rate of conversion from applicants to enrollees. When you multiply the total number of admitted students, \( x \), by 0.65, you get \( y \), the number of students who will most likely enroll. This straightforward calculation provides a powerful tool for predicting how admissions decisions translate into actual enrollees.
The admissions formula is vital for university planning and helps ensure that the freshman class sizes meet the university's capacity and goals.

Predicting Student Enrollment

Predicting the number of students who will enroll is crucial, as it impacts class size, resource allocation, and overall university planning. The linear function \( y = 0.65x \) gives us a direct method to forecast enrollment numbers.
For instance, if a university offers admission to 2000 students, we can predict how many will enroll by using the formula:

• Calculate \( y = 0.65 \times 2000 \)
• This results in \( y = 1300 \) students who are expected to enroll

These predictions are instrumental in avoiding over-enrollment or having too few students. Accurately forecasting student numbers helps in ensuring a balanced classroom experience and prevents resource strain on the university. Furthermore, using historical data to establish such percentages can be a refining tool to enhance future predictions.

Effective University Planning

University planning involves preparing for the academic year, and knowing how many students to expect is a big part of this preparation. Using linear estimates like the admissions formula allows universities to effectively plan for resources such as faculty, classroom space, and dormitory availability.
For example, if a university targets a class size of 1350, but only admits students with no consideration of conversion rates, they might under-admit or over-admit based on varying acceptance scenarios. By calculating the necessary offers of admission using \( y = 0.65x \), they estimated 2077 offers to achieve the desired class size.
This kind of strategic planning ensures that infrastructure and educational provisions align with student numbers, leading to a smoother, more efficient operation of university activities. Having reliable admissions formulas and enrollment predictions supports better resource management and helps maintain the quality of education that the institution offers.