Word Problems Involving Piecewise Functions

Summary

This educational video lesson explores real-life applications of piecewise functions. It covers scenarios like mobile plans with a base charge and additional fees for extra messages, catering services with tiered pricing based on the number of attendees, and overtime pay calculations. Each example is meticulously explained, showing how piecewise functions represent costs or payments under different conditions, providing viewers with a practical understanding of mathematical modeling in everyday situations.

Takeaways

• 📝 The lesson discusses applications of piecewise functions in real-life situations, illustrating how they can represent different pricing structures.
• 📈 A piecewise function is introduced to represent the cost of a mobile plan, where a fixed charge applies for the first 100 messages, and additional messages are charged at a per-message rate.
• 💰 The cost function for the mobile plan is defined as 300 pesos for the first 100 messages, with each message over 100 incurring an extra 1 peso charge.
• 🎉 The script provides a step-by-step explanation of how to calculate the total cost based on the number of messages sent, using the piecewise function concept.
• 🥳 Another example is given for a catering service, with different pricing tiers based on the number of attendees, from 20 persons or less to 100 or more, each with a specific cost per head.
• 🍽 The catering cost function is piecewise, starting at 150 pesos per head for 20 or fewer attendees, decreasing to 130 pesos for 21 to 50, 110 pesos for 51 to 100, and 100 pesos for over 100 attendees.
• 💼 The lesson also covers a piecewise function for a summer job's pay structure, where regular hours are paid at the normal rate and overtime hours are paid at a higher rate.
• 🕒 The weekly pay function for the summer job is detailed, with the first 40 hours paid at the standard rate and any hours beyond that at 1.5 times the normal hourly wage.
• 💡 A practical question is posed to calculate the weekly pay for a 45-hour workweek, demonstrating the application of the piecewise function to determine earnings.
• 🌐 The script concludes with an example of an internet café's pricing model, which includes a flat rate for the first hour and an additional charge for each subsequent hour.
• 💻 The mathematical model for the internet café's charges is constructed as a piecewise function, clearly outlining the cost for any given number of hours spent online.

Q & A

What is the monthly charge for the mobile plan that includes 100 free text messages?

The monthly charge for the mobile plan is 300 pesos, which includes 100 free text messages.

How much is the charge for each text message sent beyond the 100 free messages?

Each text message sent beyond the 100 free messages is charged at one peso each.

How is the amount paid by the consumer each month represented as a function of the number of messages sent?

The amount paid by the consumer each month is represented as a piecewise function where if the number of messages (m) is less than or equal to 100, the cost is 300 pesos, and if m is greater than 100, the cost is 300 pesos plus (m - 100) pesos.

What is the cost per head for catering services for a party with 20 or fewer attendees?

The cost per head for catering services for a party with 20 or fewer attendees is 150 pesos.

What is the cost per head for catering services for a party with more than 20 but up to 50 attendees?

The cost per head for catering services for a party with more than 20 but up to 50 attendees is 130 pesos.

How much does it cost per head for catering services for a party with 51 to 100 attendees?

The cost per head for catering services for a party with 51 to 100 attendees is 110 pesos.

What is the cost per head for catering services for a party with more than 100 attendees?

The cost per head for catering services for a party with more than 100 attendees is 100 pesos.

How is the weekly pay calculated for a job that pays time and a half for overtime hours worked beyond 40 hours per week?

The weekly pay is calculated as the normal hourly wage of 30 pesos times the number of hours worked up to 40, plus 1.5 times the hourly wage for each hour worked beyond 40.

What is the formula for calculating the pay for working exactly 45 hours in the job that pays overtime?

The pay for working 45 hours is 1200 pesos (30 pesos per hour for 40 hours) plus 45 pesos per hour for 5 hours of overtime, which equals 1200 + (45 * 5) pesos.

What is the flat rate for the first hour of internet surfing at the internet cafe?

The flat rate for the first hour of internet surfing at the internet cafe is 15 pesos.

How much is the additional charge for each hour of internet surfing beyond the first hour at the internet cafe?

The additional charge for each hour of internet surfing beyond the first hour is 5 pesos per hour.

How can the total charge at the internet cafe be represented as a function of the total number of hours played (x)?

The total charge at the internet cafe can be represented as a function where if x is less than or equal to 1, the charge is 15 pesos, and if x is greater than 1, the charge is 15 pesos plus 5 pesos for each additional hour (x - 1).