Answer :
Certainly! Let's carefully analyze and formulate the functions for each given real-life scenario step by step.
### 1. Expressing Total Salary as a Function of Days
Scenario: A person is earning P750.00 per day to do a certain job.
Formulate: To find the total salary [tex]\( T \)[/tex] as a function of the number of days [tex]\( x \)[/tex]:
1. Identify the daily earning rate: P750.00 per day.
2. Multiply the daily earning rate by the number of days [tex]\( x \)[/tex].
The function to represent the total salary [tex]\( T \)[/tex] is:
[tex]\[ T(x) = 750 \cdot x \][/tex]
### 2. Determining Total Savings in a Piggy Bank
Scenario: Heart started with an initial deposit of P100.00 and then deposited P15.00 each week.
Formulate: To find the total savings [tex]\( I \)[/tex] as a function of the number of weeks [tex]\( k \)[/tex]:
1. Start with the initial deposit: P100.00.
2. Add the weekly deposit of P15.00 multiplied by the number of weeks [tex]\( k \)[/tex].
The function to represent the total savings [tex]\( I \)[/tex] is:
[tex]\[ I(k) = 100 + 15 \cdot k \][/tex]
### 3. Representing Computer Rental Fee
Scenario: A computer shop charges P25.00 per hour for computer rental.
Formulate: To find the computer rental fee [tex]\( C \)[/tex] as a function of the number of hours [tex]\( h \)[/tex]:
1. Identify the hourly charge: P25.00.
2. Multiply the hourly charge by the number of hours [tex]\( h \)[/tex].
The function to represent the computer rental fee [tex]\( C \)[/tex] is:
[tex]\[ C(h) = 25 \cdot h \][/tex]
### 4. Enclosing Area with Fencing Next to a River
Scenario: Fifty meters of fencing is available to enclose a rectangular area next to a river.
Formulate: To find a function [tex]\( A \)[/tex] that represents the area enclosed in terms of [tex]\( x \)[/tex]:
1. Let [tex]\( x \)[/tex] be the length of the side perpendicular to the river.
2. Since the fencing is next to a river, only three sides are fenced: [tex]\( x \)[/tex] (length), [tex]\( x \)[/tex] (other length), and [tex]\( 50 - 2x \)[/tex] (width).
3. The area of the rectangle is then calculated by multiplying the lengths of its sides: [tex]\( x \)[/tex] and [tex]\( 50 - 2x \)[/tex].
The function to represent the enclosed area [tex]\( A \)[/tex] is:
[tex]\[ A(x) = x \cdot (50 - 2x) \][/tex]
### 5. Defining Volume of a Box Created from a Rectangle
Scenario: Squares of side [tex]\( x \)[/tex] are cut from a 7 inches by 5 inches rectangle so that its sides can be folded to create a box without a top.
Formulate: To find a function in terms of [tex]\( x \)[/tex] that defines the volume [tex]\( V \)[/tex] of the box:
1. The length and width of the rectangle are reduced by [tex]\( 2x \)[/tex] (cut from both sides).
- New length: [tex]\( 7 - 2x \)[/tex]
- New width: [tex]\( 5 - 2x \)[/tex]
2. The height of the box is the side length of the cut square: [tex]\( x \)[/tex].
3. The volume of the box is the product of its dimensions: length, width, and height.
The function to represent the volume [tex]\( V \)[/tex] of the box is:
[tex]\[ V(x) = (7 - 2x) \cdot (5 - 2x) \cdot x \][/tex]
These formulations yield the functions necessary to represent each scenario mathematically.
### 1. Expressing Total Salary as a Function of Days
Scenario: A person is earning P750.00 per day to do a certain job.
Formulate: To find the total salary [tex]\( T \)[/tex] as a function of the number of days [tex]\( x \)[/tex]:
1. Identify the daily earning rate: P750.00 per day.
2. Multiply the daily earning rate by the number of days [tex]\( x \)[/tex].
The function to represent the total salary [tex]\( T \)[/tex] is:
[tex]\[ T(x) = 750 \cdot x \][/tex]
### 2. Determining Total Savings in a Piggy Bank
Scenario: Heart started with an initial deposit of P100.00 and then deposited P15.00 each week.
Formulate: To find the total savings [tex]\( I \)[/tex] as a function of the number of weeks [tex]\( k \)[/tex]:
1. Start with the initial deposit: P100.00.
2. Add the weekly deposit of P15.00 multiplied by the number of weeks [tex]\( k \)[/tex].
The function to represent the total savings [tex]\( I \)[/tex] is:
[tex]\[ I(k) = 100 + 15 \cdot k \][/tex]
### 3. Representing Computer Rental Fee
Scenario: A computer shop charges P25.00 per hour for computer rental.
Formulate: To find the computer rental fee [tex]\( C \)[/tex] as a function of the number of hours [tex]\( h \)[/tex]:
1. Identify the hourly charge: P25.00.
2. Multiply the hourly charge by the number of hours [tex]\( h \)[/tex].
The function to represent the computer rental fee [tex]\( C \)[/tex] is:
[tex]\[ C(h) = 25 \cdot h \][/tex]
### 4. Enclosing Area with Fencing Next to a River
Scenario: Fifty meters of fencing is available to enclose a rectangular area next to a river.
Formulate: To find a function [tex]\( A \)[/tex] that represents the area enclosed in terms of [tex]\( x \)[/tex]:
1. Let [tex]\( x \)[/tex] be the length of the side perpendicular to the river.
2. Since the fencing is next to a river, only three sides are fenced: [tex]\( x \)[/tex] (length), [tex]\( x \)[/tex] (other length), and [tex]\( 50 - 2x \)[/tex] (width).
3. The area of the rectangle is then calculated by multiplying the lengths of its sides: [tex]\( x \)[/tex] and [tex]\( 50 - 2x \)[/tex].
The function to represent the enclosed area [tex]\( A \)[/tex] is:
[tex]\[ A(x) = x \cdot (50 - 2x) \][/tex]
### 5. Defining Volume of a Box Created from a Rectangle
Scenario: Squares of side [tex]\( x \)[/tex] are cut from a 7 inches by 5 inches rectangle so that its sides can be folded to create a box without a top.
Formulate: To find a function in terms of [tex]\( x \)[/tex] that defines the volume [tex]\( V \)[/tex] of the box:
1. The length and width of the rectangle are reduced by [tex]\( 2x \)[/tex] (cut from both sides).
- New length: [tex]\( 7 - 2x \)[/tex]
- New width: [tex]\( 5 - 2x \)[/tex]
2. The height of the box is the side length of the cut square: [tex]\( x \)[/tex].
3. The volume of the box is the product of its dimensions: length, width, and height.
The function to represent the volume [tex]\( V \)[/tex] of the box is:
[tex]\[ V(x) = (7 - 2x) \cdot (5 - 2x) \cdot x \][/tex]
These formulations yield the functions necessary to represent each scenario mathematically.
