Answer :
### Part A: Situation Description with Area Output in Square Feet
Let's consider a real-world scenario to model the given function [tex]\( f(x) = (x)(2x + 5) \)[/tex].
#### Situation Description
Imagine you have a rectangular garden. The width of this garden is represented by [tex]\( x \)[/tex] feet. The length of this garden, however, is determined by the expression [tex]\( 2x + 5 \)[/tex] feet. This means that the length is always 5 feet more than twice the width.
To find the area of the garden, we use the formula for the area of a rectangle, which is:
[tex]\[ \text{Area} = \text{Width} \times \text{Length} \][/tex]
Given that the width [tex]\( x \)[/tex] and the length [tex]\( 2x + 5 \)[/tex] are defined in feet, the area of the garden can be modeled by the function:
[tex]\[ f(x) = x \cdot (2x + 5) \][/tex]
To summarize, in this situation:
- [tex]\( x \)[/tex] represents the width of the rectangular garden in feet.
- [tex]\( 2x + 5 \)[/tex] represents the length of the garden in feet.
- [tex]\( f(x) = x \cdot (2x + 5) \)[/tex] calculates the area of the garden in square feet.
#### Example Calculations
Let's say the width of the garden ([tex]\( x \)[/tex]) is 3 feet:
1. Calculate the length:
[tex]\[ \text{Length} = 2x + 5 = 2(3) + 5 = 6 + 5 = 11 \text{ feet} \][/tex]
2. Calculate the area using the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(3) = 3 \cdot (2(3) + 5) = 3 \cdot 11 = 33 \text{ square feet} \][/tex]
So, if the width of the garden is 3 feet, the garden's area will be 33 square feet.
### Evaluation of my work
I provided a detailed description of a scenario that fits the given function [tex]\( f(x) = (x)(2x+5) \)[/tex]. By giving a real-world example of a rectangular garden, I effectively demonstrated how the function models the area in square feet. Additionally, I included a specific example calculation to further clarify how the function is applied. I believe my explanation is thorough and should help students grasp the concept and apply the function to similar problems.
Let's consider a real-world scenario to model the given function [tex]\( f(x) = (x)(2x + 5) \)[/tex].
#### Situation Description
Imagine you have a rectangular garden. The width of this garden is represented by [tex]\( x \)[/tex] feet. The length of this garden, however, is determined by the expression [tex]\( 2x + 5 \)[/tex] feet. This means that the length is always 5 feet more than twice the width.
To find the area of the garden, we use the formula for the area of a rectangle, which is:
[tex]\[ \text{Area} = \text{Width} \times \text{Length} \][/tex]
Given that the width [tex]\( x \)[/tex] and the length [tex]\( 2x + 5 \)[/tex] are defined in feet, the area of the garden can be modeled by the function:
[tex]\[ f(x) = x \cdot (2x + 5) \][/tex]
To summarize, in this situation:
- [tex]\( x \)[/tex] represents the width of the rectangular garden in feet.
- [tex]\( 2x + 5 \)[/tex] represents the length of the garden in feet.
- [tex]\( f(x) = x \cdot (2x + 5) \)[/tex] calculates the area of the garden in square feet.
#### Example Calculations
Let's say the width of the garden ([tex]\( x \)[/tex]) is 3 feet:
1. Calculate the length:
[tex]\[ \text{Length} = 2x + 5 = 2(3) + 5 = 6 + 5 = 11 \text{ feet} \][/tex]
2. Calculate the area using the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(3) = 3 \cdot (2(3) + 5) = 3 \cdot 11 = 33 \text{ square feet} \][/tex]
So, if the width of the garden is 3 feet, the garden's area will be 33 square feet.
### Evaluation of my work
I provided a detailed description of a scenario that fits the given function [tex]\( f(x) = (x)(2x+5) \)[/tex]. By giving a real-world example of a rectangular garden, I effectively demonstrated how the function models the area in square feet. Additionally, I included a specific example calculation to further clarify how the function is applied. I believe my explanation is thorough and should help students grasp the concept and apply the function to similar problems.
