Brayden has 24 feet of fence available to build a rectangular fenced in area. If the width of the rectangle is xx feet, then the length would be one half, left parenthesis, 24, minus, 2, x, right parenthesis, . 
2
1
​
 (24−2x). A function to find the area, in square feet, of the fenced in rectangle with width xx is given by f, of, x, equals, one half, x, left parenthesis, 24, minus, 2, x, right parenthesis, .f(x)= 
2
1
​
 x(24−2x). Find and interpret the given function values and determine an appropriate domain for the function.
Answer
Attempt 1 out of 2

f, of, minus, 3, equalsf(−3)= 
, meaning when the width of the rectangular area is 
 ft, commaft, the area would be 
 ft, squared, .ft 
2
 . This interpretation 
 in the context of the problem.
f, of, 5, point, 5, equalsf(5.5)= 
, meaning when the width of the rectangular area is 
 ft, commaft, the area would be 
 ft, squared, .ft 
2
 . This interpretation 
 in the context of the problem.
f, of, 15, equalsf(15)= 
, meaning when the width of the rectangular area is 
 ft, commaft, the area would be 
 ft, squared, .ft 
2
 . This interpretation 
 in the context of the problem.
Based on the observations above, it is clear that an appropriate domain for the function is 
.

### Step 1: Define the function for area

The function for the area \( f(x) \) of the rectangular fenced area is given by:

\[
f(x) = \frac{1}{2} x (24 - 2x)
\]

Where:
- \( x \) is the width of the rectangle (in feet),
- \( 24 - 2x \) is the expression for the remaining length of the rectangle.

### Step 2: Interpret the given function values

We will substitute the given values of \( x \) (width) into the area function and interpret the results.

1. **When \( x = -3 \):**
   \[
   f(-3) = \frac{1}{2} (-3) (24 - 2(-3)) = \frac{1}{2} (-3) (24 + 6) = \frac{1}{2} (-3) (30) = \frac{1}{2} (-90) = -45
   \]
   This result doesn't make sense in the real-world context because a negative area is impossible. So, this is not a valid solution. When \( x = -3 \), it is outside the valid domain of the function, and the interpretation would be that the width cannot be negative.

2. **When \( x = 5.5 \):**
   \[
   f(5.5) = \frac{1}{2} (5.5) (24 - 2(5.5)) = \frac{1}{2} (5.5) (24 - 11) = \frac{1}{2} (5.5) (13) = \frac{1}{2} (71.5) = 35.75
   \]
   This means that when the width of the rectangular area is 5.5 feet, the area would be 35.75 square feet.

3. **When \( x = 15 \):**
   \[
   f(15) = \frac{1}{2} (15) (24 - 2(15)) = \frac{1}{2} (15) (24 - 30) = \frac{1}{2} (15) (-6) = \frac{1}{2} (-90) = -45
   \]
   Again, we get a negative area of \(-45\), which is not physically possible, meaning that \( x = 15 \) is also outside the valid domain for this problem.

### Step 3: Determine the appropriate domain

To find the domain, we need to ensure both the width and the length of the rectangle are non-negative and make sense in the context of the problem.

From the expression for the length \( 24 - 2x \), we require:
\[
24 - 2x \geq 0 \quad \Rightarrow \quad x \leq 12
\]
Also, since the width \( x \) cannot be negative:
\[
x \geq 0
\]
Therefore, the domain of the function is:
\[
0 \leq x \leq 12
\]

### Conclusion

- The appropriate domain for the function is \( 0 \leq x \leq 12 \).
- \( f(-3) \) results in a negative area, which is not valid in the real-world context.
- \( f(5.5) = 35.75 \) square feet means that when the width is 5.5 feet, the area is 35.75 square feet.
- \( f(15) \) also results in a negative area, which is not valid.

### 5 Follow-up questions:
1. What would be the maximum possible area for this rectangular fence?
2. Can you graph the function \( f(x) = \frac{1}{2} x (24 - 2x) \) and identify its critical points?
3. How does the area change as \( x \) increases within the valid domain?
4. What would the length be when \( x = 6 \) feet?
5. What is the width of the rectangle when the area is maximized?

### Tip:
Always check the physical constraints of a word problem, like ensuring non-negative lengths and areas, to help define the correct domain for the function.

Brayden has 24 feet of fence available to build a rectangular fenced in area. If the width of the rectangle is x feet, then the length would be one half (24 − 2x). A function to find the area, in square feet, of the fenced-in rectangle with width x is given by f(x) = (1/2) x(24 − 2x). Find and interpret the given function values and determine an appropriate domain for the function.

Solve for the area of a rectangular fenced region using 24 feet of material. Explore the function f(x) = (1/2) x(24 − 2x) and interpret function values with a detailed step-by-step explanation. Suitable for students in Grades 9-10 studying quadratic functions and real-world applications.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation