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
Olivia looked at her dad, feeling a mix of relief and frustration.
Olivia- I can't believe I didn't see it right away. It was actually pretty straightforward once I got it. I guess I was just overthinking it.
Tim- Well, sometimes it takes a moment to click. You figured it out and explained it really well. Good job!
Olivia smiled at her dad's encouragement, but noticed his slightly puzzled expression.
Olivia- Are you sure you're following all of this?
Tim- (scratching his head) I'm trying, but you're going pretty fast. Exponential growth and natural logs... not exactly my strong suit. I'm better with languages.
Olivia noticed her dad's slightly bewildered expression and chuckled.
Olivia- Oh, I see. Well, I'm actually pretty good with languages too, but I just like math more. It's like solving puzzles for me.
Tim- That makes sense. You seem to have a knack for it. But are you sure you've solved the task correctly? Sometimes I think I'm following, but I might be missing something.
Olivia- I don't do wrong.
Tim chuckled at Olivia's confident remark.
Tim- I love that confidence. It's one of the things I admire about you.
Olivia- Thanks, Dad.
Olivia quickly cleared her homework and packed away her books. Tim finished tidying up the kitchen and then they both made their way to the living room.
Tim- Ready to dive into another episode?
Olivia- Absolutely! I've been looking forward to this all day.
Tim- Same here. It's nice to have this downtime after a busy day.
They snuggled into their seats, the soft glow of the TV screen lighting up the room as the opening credits of their favorite series started.
As they watched, they laughed and chatted about the plot twists and character developments. The episode drew them in, and they were soon engrossed in the story. As the episode wrapped up, Olivia stretched and yawned.
Olivia- That was a great episode.
Tim- Yes it was great and now it's time for you to go to bed.
They both got up from the couch, and Tim followed Olivia up the stairs. Olivia, always a few steps ahead, made her way up while Tim trailed behind, ready for a bit of playful mischief. (It's his duty as her dad.)
When they reached the top of the stairs, Tim couldn't resist the urge to tease her. He snuck up behind Olivia and gently grabbed her by the sides.
Olivia- Hey! What are you—
Before she could react, Tim had lifted her off her feet and swung her around. With a playful laugh, he turned her upside down, making her squeal with laughter.
Olivia- (giggling) Dad! Put me down!
Tim- Never!
With a final playful flourish, Tim carefully threw Olivia on her bed, her laughter still echoing in the room. He ruffled her hair affectionately.
Olivia- (sitting up and catching her breath) You're the worst.
Tim- And you love it. You've had a busy day. Get some rest.
Olivia- (grinning) Thanks, Dad. Goodnight!
Tim- Goodnight, Liv. Sweet dreams.
Tim headed to his room, still smiling from the playful interaction with Olivia. He took a quick glance at his phone as he entered, seeing a new notification from Angela. Opening the message, he saw an invitation to a grill party this weekend—a casual get-together with friends.
Tim quickly typed a response, letting Angela know he was looking forward to it.
After sending the message, Tim put his phone aside and headed for the bathroom. He took a long, hot shower, letting the warmth relax his muscles and clear his mind. As he stood under the stream of water, his thoughts wandered. He found himself just staring at the wall, lost in thought, as the steam enveloped him.
When he finally finished his shower, Tim dried himself off with a towel, put on some boxers and comfortable sweatpants, and brushed his teeth. His mind continued to drift, reflecting on a lot of different things.
Once he was ready for bed, he climbed under the covers, feeling the soft embrace of his bed. He settled in, trying to quiet his thoughts and unwind. He glanced at the clock—early, but not too early to wind down.
As the days passed, Tim and Olivia settled into their routines. Thursday arrived with the familiar rhythm of their lives. Tim and Lucy were out on patrol, working together seamlessly as they navigated the streets. The morning sun cast a golden glow over the city, and the radio chatter kept them updated on various calls.
In a lull between assignments, Lucy glanced over at Tim, her curiosity getting the better of her.
Lucy- So, how's Olivia doing? I haven't had a chance to catch up with her lately.
Tim- Oh, she's doing great. Busy with school and practice, as usual. Why do you ask?
Lucy hesitated for a moment, trying to come up with a plausible reason.
Lucy- Oh, no particular reason. Just curious, you know. I like to keep up with how everyone's doing.
Tim gave her a curious look but seemed satisfied with the explanation.
Tim- Fair enough.
Lucy smiled, and they continued their patrol, chatting about the day's events and other topics. Though Lucy had wanted to get a sense of Olivia's recent gossip to share with her, she was relieved that Tim didn't press further. The conversation drifted to other subjects, and the morning passed in comfortable companionship.
As the patrol continued, Lucy's curiosity about Olivia remained. She decided to try a different approach to get a bit more information.
Lucy- So, Tim, Olivia is coming to the station tomorrow, right?
Tim glanced at her, considering her question for a moment.
Tim- Yeah. How did you know?
Lucy quickly thought of a way to steer the conversation in a different direction, not wanting to reveal that Olivia had told her directly.
Lucy- Oh, you know, I just think I heard she'd be there. Anyway, have you noticed how nice the weather's been lately?
Tim looked at her, clearly aware of the attempt to change the subject but decided to go along with it.
They continued driving, discussing the weather. Lucy felt a sense of relief, glad that Tim hadn't pressed her for more details about Olivia's visit.
As they rounded a corner, they noticed a small gathering of people at a local park. Tim slowed the car, and they both scanned the area for any signs of trouble. Seeing nothing out of the ordinary, they continued their patrol, the conversation flowing easily between them.
Lucy- So, any plans for the weekend?
Tim- Well, besides the grill party at Angela's, probably just hanging out with Olivia.
Lucy- That sounds great.
Tim- Yeah, it is.
After a few moments of silence, Tim glanced at Lucy, a thoughtful expression on his face.
Tim- Can I ask you something?
Lucy- Sure, what's up?
Tim- What's the deal with you and Olivia lately? You two seem to be getting closer, and she's mentioned you a few times at home.
Tim hesitated for a moment, choosing his words carefully.
Tim- I know you've had some conversations that she's kept a bit private. She has never been like that. Is everything okay?
Lucy shifted slightly in her seat, trying to keep her composure as she considered how to respond. She could feel Tim's concern and knew she had to tread carefully.
Lucy- Oh, everything's perfectly fine, Tim. Olivia and I have just been talking about some of her interests and things going on at school. It's nothing major.
Tim- I see. But if everything's fine, why isn't she sharing more with me? I've noticed she's been a bit more reserved lately about her conversations with you.
Lucy- Well, teenagers often have phases where they feel like they need to keep certain things to themselves. It's not that she's hiding anything from you. Sometimes they just need to talk to someone who isn't directly involved in their daily lives. It's a normal part of growing up.
Tim glanced at Lucy, his concern evident.
Tim- I understand that. But Olivia just turned 11. It seems a bit early for her to start having these kinds of phases.
Lucy shifted uncomfortably in her seat, feeling the weight of Tim's concern. She knew she needed to address his worries while keeping the truth concealed.
Lucy- You're right. Maybe we weren't talking just about her.
Her voice peaked slightly. Tim looked at her with a confused expression.
Lucy- We had a few conversations about some things that weren't directly related to her. I might have accidentally told her a little too much about me.
Tim- So she's your therapist?
Lucy- No, no. She was talking about a girl at school and a few things just slipped.
Tim's expression shifted from concern to amusement as he chuckled softly.
Tim- I thought there was something going on with Olivia! I'm glad it's nothing serious.
Lucy- No, nothing like that. Just some casual conversations that went off-topic.
Tim- Now I'm even more curious. What kind of things did you end up sharing with her?
Lucy- Oh, it's nothing you need to worry about. Just some... you don't need to know.
Tim- Now you've got me even more interested! What's the scoop?
Lucy- Really, Tim, it's nothing.
Tim- Alright, alright. I'll let it go— for now.
Lucy- For ever!
Tim- Fine.
The next day, as Tim and Lucy continued their patrol, the rhythm of their workday unfolded seamlessly. Around 1 PM, it was time for Tim to pick up Olivia from school.
Tim parked the car outside Olivia's school, scanning the crowd of kids emerging from the building. When he spotted Olivia, she waved enthusiastically and hurried over to the car, her backpack bouncing with each step.
Olivia- Hey, Dad!
Tim- Hey, Liv. Ready for lunch?
Olivia- Absolutely! I'm starving.
They drove over to the food trucks where Tim's colleagues—Angela, Nyla, John, Lucy, Aaron already gathered. Olivia eagerly selected her lunch from the taco truck and the joined her dad's friends, while Tim ordered for himself.
Tim collected their food and walked over to the tables. Olivia was already seated with the others, her eyes scanning her haul with anticipation.
Olivia wasted no time and took a bite of her tacos, but she quickly noticed the enticing aroma of the BBQ chicken Tim had ordered.
Olivia- Dad?
Tim knew what she was about to ask, so he handed over a piece of the chicken, and Olivia's eyes widened with delight as she tasted it.
Without hesitation, Olivia swapped her tacos for Tim's BBQ chicken.
Tim- Well, I guess I'm having tacos today.
Aaron- Man, Bradford, you just take what you get, don't you?
Olivia- Of course he does.
Lucy- She's got him wrapped around her finger.
Olivia- Maybe a little. But he doesn't mind, right, Dad?
Tim- Not at all. As long as you're happy.
John- I think Tim knows who's really in charge here.
The group laughed and enjoyed their meal, chatting and sharing stories. After lunch, Tim and Lucy prepared to head back out on patrol. Tim dropped Olivia off at the station before they left.
Tim- Alright, Liv. Don't get into too much trouble while I'm out.
Olivia- I'll be good, Dad. Promise.
Tim and Lucy resumed their patrol, the afternoon sun casting long shadows as they drove through the city streets. Later, when they headed back to the station to do paperwork, Lucy went straight to Olivia.
Lucy- Hey, Liv. Quick heads up—your dad was a bit concerned yesterday. He thought you were keeping things from him. I told him we were talking about my stuff, just to make sure we're on the same page if he asks.
Olivia- Got it. No worries.
Lucy- Great. Now, spill the tea. What's the latest gossip at the station?
Olivia's eyes lit up, and she leaned in closer, ready to share all the juicy details.
Olivia- Well, you didn't hear this from me, but...
Olivia glanced around to make sure no one else was listening before continuing in a hushed voice.
Olivia- So, apparently, Angela and Nyla were talking about this new guy in the detective unit. Rumor has it he's got a reputation for being a bit of a ladies' man. They've been placing bets on how long it will take before he tries to ask someone out at the station.
Lucy- No way! Who do you think he's going to ask out first?
Olivia- Angela thinks it'll be Celina, but Nyla is convinced he's going to try his luck with Grey's assistant. You know, the one who just started last month.
Lucy- That's wild. Anything else?
Olivia- Well, there's a rumor that Travis might be planning to propose to his girlfriend soon. He's been acting really secretive lately, and some of the officers think he's been sneaking around to plan the perfect proposal.
Lucy- Witch one?
Olivia- The one with the black hair. I think his name is Harrison.
Lucy- That's adorable! I hope it's true.
Olivia- Me too! And, oh, you'll love this—apparently, someone found a stash of donuts hidden in the break room ceiling. No one knows who put them there.
Lucy- Smitty
Olivia- I know. I told them that only Smitty would do such thing. But they think it wasn't him because it would be too much work to get them up there.
Olivia leaned in even closer, her eyes sparkling with excitement as she continued to spill the latest gossip. Lucy listened intently, enjoying every bit of the inside scoop Olivia shared. From rumors of secret office romances to speculation about upcoming promotions, Olivia had all the details.
Eventually, Lucy glanced at the clock and realized she needed to get back to her paperwork.
Lucy- This has been great, Liv. Thanks for filling me in. But I should probably get back to work before your dad comes to get the paperwork.
Olivia- No problem. Anytime you need the latest, you know where to find me. Just make sure my dad doesn't find out I'm your source.
Lucy- Your secret's safe with me.
Lucy smiled, feeling a renewed sense of camaraderie with Olivia. As Olivia walked back to the break room, she couldn't help but chuckle at some of the stories. The station was always full of surprises, and having Olivia as her unofficial source made the workday just a bit more entertaining.
The rest of the day passed without incident, and soon enough, Tim and Lucy were wrapping up their shift. The station was starting to quiet down as the evening crew began to take over.
Tim- Ready to head out?
Lucy- Absolutely. Just need to grab a few things from my desk.
As Lucy gathered her belongings, Tim made his way to the break room to find Olivia. He found her chatting with a couple of officers, her usual bubbly self.
Tim- Hey, Liv. Ready to go?
Olivia- Yep!
She said her goodbyes and grabbed her backpack, joining her dad as they headed towards the exit. Lucy caught up with them, a friendly smile on her face.
Lucy- See you tomorrow, Tim. Bye, Olivia!
Olivia- Bye, Lucy! See you!
Tim and Olivia walked out of the station, the evening air cool and refreshing after the long day. They made their way to the car, the comfortable routine of their evening commute settling in.