Questions: Problem. 14: A certain small college's student enrollment is growing at the rate of P'(t) = 2900 / ((1/5 t + 1)^(3/2)) students per year, t years from now. If the current student enrollment is 1200, find an expression giving the enrollment t years from now. What will the enrollment be 5 years from now (round your answer to the nearest student).

Problem. 14: A certain small college's student enrollment is growing at the rate of
P'(t) = 2900 / ((1/5 t + 1)^(3/2))
students per year, t years from now. If the current student enrollment is 1200, find an expression giving the enrollment t years from now.
What will the enrollment be 5 years from now (round your answer to the nearest student).

Transcript text: Problem. 14: A certain small college's student enrollment is growing at the rate of \[ P^{\prime}(t)=\frac{2900}{\left(\frac{1}{5} t+1\right)^{\frac{3}{2}}} \] students per year, $t$ years from now. If the current student enrollment is 1200 , find an expression giving the enrollment $t$ years from now. What will the enrollment be 5 years from now (round your answer to the nearest student).

Solution

Solution Steps

To solve this problem, we need to integrate the given rate of change of student enrollment to find the total enrollment function $ P(t) $. Then, we will use the initial condition to find the constant of integration. Finally, we will evaluate the enrollment function at $ t = 5 $ to find the enrollment 5 years from now.

Solution Approach

Integrate the given rate of change $ P'(t) $ to find $ P(t) $.
Use the initial condition $ P(0) = 1200 $ to find the constant of integration.
Evaluate $ P(t) $ at $ t = 5 $ to find the enrollment 5 years from now.

Step 1: Integrate the Rate of Change Function

Given the rate of change of student enrollment: \[ P'(t) = \frac{2900}{\left(\frac{1}{5} t + 1\right)^{3/2}} \] we integrate $ P'(t) $ with respect to $ t $ to find the enrollment function $ P(t) $: \[ P(t) = \int \frac{2900}{\left(\frac{1}{5} t + 1\right)^{3/2}} \, dt \] The integral of $ P'(t) $ is: \[ P(t) = 30200 - \frac{29000}{\left(\frac{1}{5} t + 1\right)^{1/2}} + C \]

Step 2: Apply the Initial Condition

We are given the initial condition $ P(0) = 1200 $. Substituting $ t = 0 $ into the enrollment function: \[ 1200 = 30200 - \frac{29000}{\left(\frac{1}{5} \cdot 0 + 1\right)^{1/2}} + C \] Simplifying, we find: \[ 1200 = 30200 - 29000 + C \] \[ C = 1200 - 1200 = 0 \]

Step 3: Substitute the Constant Back into the Enrollment Function

Substituting $ C = 0 $ back into the enrollment function: \[ P(t) = 30200 - \frac{29000}{\left(\frac{1}{5} t + 1\right)^{1/2}} \]

Step 4: Evaluate the Enrollment Function at $ t = 5 $

To find the enrollment 5 years from now, we evaluate $ P(t) $ at $ t = 5 $: \[ P(5) = 30200 - \frac{29000}{\left(\frac{1}{5} \cdot 5 + 1\right)^{1/2}} \] \[ P(5) = 30200 - \frac{29000}{\left(1 + 1\right)^{1/2}} \] \[ P(5) = 30200 - \frac{29000}{\sqrt{2}} \] \[ P(5) \approx 30200 - 20406.0967 \] \[ P(5) \approx 9693.9033 \]

Final Answer

$\boxed{P(5) \approx 9694}$

Was this solution helpful?

Unhelpful

Helpful

Questions asked by other users

< Previous Next >

Thank you for your feedback.

Copied!