What if Tim had a daughter?

Part 9

At Tim's house, the evening was settling into its usual comforting routine. After pulling into the driveway, Tim and Olivia made their way inside, greeted by the familiar warmth of home.
Olivia- Dad, can we order pizza tonight?
Tim chuckled at her enthusiasm but then thought about their recent streak of takeout meals.
Tim- You know, Liv, we've been ordering out a lot lately. How about we cook something together tonight instead? It'll be healthier, and I could use a break from all the fast food.
Olivia- Hmm, okay. What should we make?
Tim- How about we make some homemade chicken tenders with a side of sweet potato fries and a salad? It's healthy and delicious.
Olivia- That sounds good! I can help with the salad.
Tim smiled at her willingness and led the way to the kitchen. He gathered the ingredients for their impromptu dinner: chicken breasts, breadcrumbs, eggs, sweet potatoes, and a variety of salad greens and veggies. Olivia started washing the vegetables.
Tim- So, you start on the salad while I get the chicken tenders and fries ready.
Olivia- Got it, chef!
They both laughed, enjoying the playful atmosphere. Tim cut the chicken breasts into strips and set up a breading station with flour, beaten eggs, and seasoned breadcrumbs. Olivia was busy chopping cucumbers, tomatoes, and carrots for the salad.
Tim- You're getting really good at that, Liv. Maybe you'll be a chef someday.
Olivia- Maybe! Or maybe I'll just be really good at making salads.
Tim chuckled, turning his attention back to the stove. He coated the chicken strips in flour, dipped them in egg, and rolled them in breadcrumbs before heating up a skillet on the stove.
Next, he cut the sweet potatoes into fries, tossed them with a bit of olive oil and seasoning, and spread them out a baking sheet.
Tim- Alright, into the oven they go.
As the chicken tenders are sizzling in the pan and sweet potato fries baked, Olivia finished preparing the salad. They added a mix of greens, chopped veggies, and added Salad dressing.
Tim smiled, feeling a sense of contentment. When the chicken tenders were golden brown and the sweet potato fries were crispy, he pulled them out of the oven.
Tim- Dinner is served!
Olivia- Looks awesome, Dad. Thanks.
They sat down to eat, savoring the fruits of their labor. The meal was simple but satisfying, a welcome change from their usual takeout routine.
Tim- This is nice, isn't it? We should do this more often.
Olivia- Yeah, I like cooking with you. Plus, this tastes way better than pizza.
Tim smiled, feeling a sense of contentment. After dinner, he cleaned up the kitchen while Olivia did her homework at the kitchen counter. She worked diligently through her assignments, but when she reached the fifth task, it didn't make any sense. Frustrated, she skipped it for now and finished the rest.
After Tim was done cleaning, Olivia called him over.
Olivia- Dad, can you help me with this? It doesn't make any sense.

Math Problem:

Consider an exponential growth function that describes the population of a species. The population P(t) at time t in years is given by the equation:
P(t) = P0 • e^kt

where P0 is the initial population, e is the base of the natural logarithm, k is the growth rate, and t is time in years.

You are given the following information:

The population doubles every 5 years.
At time
1. t=0, the population is
2. P0= 500.

Task:
1. Find the growth rate k based on the given information.
2. Use the growth rate k to determine the population after 10 years.

Tim read the problem slowly, trying to make sense of it. Olivia, who had been quietly working on it, suddenly brightened up.
Olivia- (reading aloud) "The population doubles every 5 years." That means that in 5 years, the population is twice what it was initially. So we need to find the growth rate k first!
She stopped mid-sentence, and a realization dawned on her.
Olivia- Oh, I get it now! It's actually pretty simple!
Tim looked at her, intrigued but not fully grasping it yet.
Tim- (trying to follow) Really? What's the trick?
Olivia- Okay, so if the population doubles every 5 years, that means after 5 years, the population will be 2⋅P0. In our case, that's 2⋅500=1000.
To find k, we use the formula: 1000=500⋅e^5k
We can simplify this to:
2=e^5k
To solve for k, we take the natural logarithm of both sides:
ln(2)=5k
k=ln(2)/5
Then, to find the population after 10 years, we plug k back into the formula:
P(10)=500⋅e^10k
Since e^10k is the same as (e^5k)^2, and e^5k=2:
e^10k = 2^2 = 4
P(10)=500⋅4=2000