Offering Help

Michael was stuck at the counter, trying not to stare as the owner, Mrs. Estolas, chatted with the manager. She had that same wow factor as Catherine, looking like she could be on a magazine cover instead of in a restaurant. He was still kicking himself for getting caught with his phone out. He needed to stay sharp now; with the big boss around, one more slip-up could mean his job. 

Moments later, the customers that he had served earlier were asking for the bill. Michael acknowledged their request, quickly processing their bill. He kept his focus, determined not to mess up again, especially with Mrs. Estolas in the restaurant. Handing over the receipt, he flashed a professional smile, thanking them for dining at Racks. The customers left, and Michael turned to clean the table, keeping an eye on the door for new arrivals. 

As he was doing that, he cast a side glance at Catherine, because who wouldn't? Her face alone was charming and attractive, if not for her ice-princess-like facade. 

Catherine seemed to have noticed him staring as she glanced at him. Michael quickly looked away from Catherine, feeling a bit embarrassed for getting caught staring. Meanwhile, the manager approached him and led him to a quieter spot where they could speak one-on-one. 

Once they were at the spot, the manager asked.

"What are you doing? Using a phone at work?" 

"I'm sorry sir, I was studying for my assessment. My professor has offered me a chance to join a competition in the PSME. I have to be prepared." 

"I understand that but you know the rules, I don't mind you students studying as long as it doesn't interfere with your work. It's a good thing that it was Catherine who saw you, not her mother." 

"I'm really sorry, manager. It will never happen again, I swear." 

"It better be. Now I'll talk to Mrs Estolas, you handle her daughter." 

"Wait, me? Are you sure about that sir? Because I already left a bad impression on her. It's going to be awkward."

"Well, it's your job to make sure our customers are satisfied, Michael, and that includes Ms. Catherine Estolas," the manager said firmly. "Just be professional, offer her the best service, and try to make a good impression this time. We can't afford to upset the Estolas family."

Michael nodded. He straightened his uniform and took a deep breath, mentally preparing himself to approach Catherine. He waited for the manager to escort the mother to his office before making his move.

As he was about to head to her table, he noticed Catherine pulling something out from her bag. It turns out to be a textbook of some kind and a yellow pad paper. Is she planning on studying here? 

He doesn't know much about her other than her family who are fully loaded, owns a lot of real-estate properties, and have business in every major city in Metro Manila. 

Well, what is his business to know about her anyway? 

Approaching Catherine's table, he cleared his throat. Despite him making that sound, Catherine didn't bat an eye, instead, she opened her textbook which seemed to be a mathematics one, differential equation to be exact. Is she in the engineering field perhaps?

"Good afternoon ma'am, can I take your order?" 

"Did I ask you to come already?" Catherine said without looking up.

Michael felt a bit taken aback by her blunt response but managed to maintain his composure. "My apologies, ma'am. I just wanted to check if you needed anything," he replied, trying to sound as professional as possible.

Catherine finally looked up from her textbook, her expression still nonchalant. "Just a coffee, black, no sugar," she said, returning her gaze to the book.

"Right away, ma'am," Michael said and quickly turned to place her order.

Returning with her coffee, he placed it gently on the table. "Here's your coffee, ma'am. If there's anything else you need, please let me know," he said, trying to engage her in conversation.

Catherine glanced at the coffee and then back at her book. "Thank you," she said curtly, not giving any further indication of wanting to talk.

Feeling a bit awkward, Michael decided not to push further. He took a few steps back, still keeping an eye on her table in case she needed anything else.

As time flew by, Michael began to notice her struggling. He checked up on her again. 

"Ma'am?" 

"Sir…whoever you are, please do not come to my table when I'm not calling for you. Don't you see I'm busy here?" 

"I'm sorry ma'am, but I can't help but notice what you are working on. Forgive me for asking but are you an engineering student or a math student?" 

Catherine sighed. "I'm a chemical engineering student. Now, does that satisfy you? Can you leave me alone now?"

"Well…I was wondering if I could offer you some help. You know, I am quite confident that I aced my differential equation midterms." 

Catherine raised an eyebrow, her expression shifting from annoyance to mild surprise. "You? A waiter, offering to help me with differential equations?"

Michael felt a bit awkward but stood his ground. "Yes, ma'am. I'm actually a mechanical engineering student working part-time. I've got a good grasp of differential equations. Maybe I could help you out?"

There was a brief silence as Catherine assessed him, her icy demeanor thawing slightly. "Fine," she finally said, "show me how you would solve this problem." She pointed to a complex equation on her paper. "If you can't solve it, don't bother me, it looks to me like you are just acting cool."

Michael ignored her words and simply focused on the equation. 

In a continuously stirred tank reactor (CSTR), two reactants A and B react to form a product. The reaction rate R is defined as R = k * (C_A)^2 * C_B, where C_A and C_B are the concentrations of reactants A and B, and k is the reaction rate constant. The reactor receives a continuous supply of reactant A at concentration C_A0 and flow rate F. Reactant B is present in the reactor from the beginning. The volume of the reactor is V. Assuming perfect mixing and constant volume, derive the differential equation describing the change in concentration of reactant A over time.

Given Parameters:

Inflow concentration of A (C_A0)

Rate constant (k)

Flow rate (F)

Reactor volume (V)

The initial concentration of B in the reactor (C_B0)

Objective:

Formulate the differential equation for the concentration of reactant A over time. This equation should account for inflow, outflow, and the reaction rate within the CSTR. 

After reading the problem, Michael now understands why Catherine is having trouble. It was a really difficult problem. But there's this weird sensation in his brain again, as if his brain is understanding what needs to be done. 

"So? Can you solve it?"

"I can," Michael said, leaning forward, and began pointing fingers to the text. "The first thing we need to do is analyze the inflow and outflow of reactant A. Since the volume is constant, the inflow rate of A is F * C_A0. For the outflow, it's F times the concentration of A in the reactor, which is simply F * C_A. Now, the reaction rate is k * (C_A)^2 * C_B. To find the rate of change of concentration of A, we set up a differential equation using these terms."

Catherine was watching him closely, her expression turning from skepticism to interest.

Michael continued, "So, the differential equation is dC_A/dt = F/V * (C_A0 - C_A) - k * (C_A)^2 * C_B. This equation describes how the concentration of A changes over time considering the inflow, outflow, and reaction rate."

Catherine looked surprised at the simplicity and clarity with which he explained it. "That... actually makes sense," she admitted. 

"Yep, it's all about breaking down the problem and understanding each component," Michael said with a small smile, happy to have been able to help.

Catherine nodded, a hint of appreciation showing in her eyes. "Thank you... I didn't expect that from you."

"No problem, ma'am. Anything else I can assist with?" Michael asked, feeling a bit more confident now.

"Uhm…there's still three more I have to solve…Of course, I won't ask you to solve them for me…is it okay…if you can help me with those questions as well?" Catherine stammered, her cheeks flushing slightly. 

Hoh? Why is she acting so bashful now? 

Michael rubbed the back of his head. "I could ma'am, but I'm in the middle of my duty." 

"Is that so…" Catherine mumbled. "Then how about later? After your work?" 

"Eh? Are you serious ma'am? I don't know because my work ends at ten o'clock in the evening…" 

"I'll wait, and I'll pay for the troubles. I really need to finish this." 

"How much are we talking here?" Michael asked. When the opportunity strikes, you have to grab it with your hand. 

"Is three thousand good for you?" 

"Deal!" Michael replied without hesitation. Three thousand pesos is like seven days of his salary.

"Okay, add me on Messenger so we can chat later," Catherine said, pulling out her phone. 

"What's your name?" Michael asked, pulling his phone out also. 

"Catherine Estolas…my profile pic is me sitting on a bench."

Michael searched her name up and looked for that exact profile picture. And there he saw it, even in her profile pic, she is stunningly beautiful. To think she'd let him add her is surprising. He quickly sent a friend request, which she accepted almost immediately.

"Alright, I've added you," Michael said, trying to hide his surprise.

"Good, see you later then."