Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Title: Understanding Enrollment Trends with Mathematical Models**
In this exercise, we explore a mathematical model used to represent the percentage of female enrollment over time at Dogwood University. The function given is:
\[ P(t) = \frac{50}{1 + 9e^{-0.3t}} \quad (0 \leq t \leq 40) \]
Here, \( t = 0 \) represents the school year starting in 1980, and \( t \) is measured in years. The function \( P \) provides us with the percentage directly, eliminating the need for conversion from a decimal.
**Questions:**
(a) **Initial Enrollment Percentage in 1980:**
According to this model, what percentage of the total enrollment was female during the school year starting in 1980? Calculate \( P(0) \) to find this percentage. If necessary, round to the nearest tenth of a percent.
(b) **Long-Term Female Enrollment Percentage:**
If the model continues to represent the trend at Dogwood University in the future, what will the percentage of female students be in the long run? Calculate the limit of \( P(t) \) as \( t \) approaches infinity. Round your answer to the nearest tenth of a percent if necessary.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7bdec363-0864-48e9-9cb3-5716963e6c1f%2F398a9426-1e84-4fca-80e9-d2a6b56ee8bf%2F7hob3a5.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Understanding Enrollment Trends with Mathematical Models**
In this exercise, we explore a mathematical model used to represent the percentage of female enrollment over time at Dogwood University. The function given is:
\[ P(t) = \frac{50}{1 + 9e^{-0.3t}} \quad (0 \leq t \leq 40) \]
Here, \( t = 0 \) represents the school year starting in 1980, and \( t \) is measured in years. The function \( P \) provides us with the percentage directly, eliminating the need for conversion from a decimal.
**Questions:**
(a) **Initial Enrollment Percentage in 1980:**
According to this model, what percentage of the total enrollment was female during the school year starting in 1980? Calculate \( P(0) \) to find this percentage. If necessary, round to the nearest tenth of a percent.
(b) **Long-Term Female Enrollment Percentage:**
If the model continues to represent the trend at Dogwood University in the future, what will the percentage of female students be in the long run? Calculate the limit of \( P(t) \) as \( t \) approaches infinity. Round your answer to the nearest tenth of a percent if necessary.
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