Answer :
To determine which function models the monthly profit for [tex]\( x \)[/tex] trinkets produced, we need to evaluate the given functions at the provided data points and see which one matches the profit values.
The points given are:
[tex]\( (0, -3125) \)[/tex]
[tex]\( (50, 0) \)[/tex]
[tex]\( (100, 1875) \)[/tex]
[tex]\( (150, 2500) \)[/tex]
[tex]\( (200, 1875) \)[/tex]
[tex]\( (250, 0) \)[/tex]
We are provided with the following potential profit functions:
1. [tex]\( f(x) = -4(x-50)(x-250) \)[/tex]
2. [tex]\( f(x) = -\frac{1}{4}(x-50)(x-250) \)[/tex]
3. [tex]\( f(x) = 28(x+50)(x+250) \)[/tex]
4. [tex]\( f(x) = \frac{1}{28}(x+50)(x+250) \)[/tex]
We need to find out which one matches the provided points table.
For each function, we evaluate it at [tex]\( x = 0, 50, 100, 150, 200, 250 \)[/tex] to see if the function values [tex]\( y \)[/tex] match.
### Evaluating the Functions:
1. [tex]\( f(x) = -4(x-50)(x-250) \)[/tex]
- [tex]\( f(0) = -4(0-50)(0-250) = -4(-50)(-250) = -50000 \)[/tex]
- [tex]\( f(50) = -4(50-50)(50-250) = -4(0)(-200) = 0 \)[/tex]
- [tex]\( f(100) = -4(100-50)(100-250) = -4(50)(-150) = 30000 \)[/tex]
- [tex]\( f(150) = -4(150-50)(150-250) = -4(100)(-100) = 40000 \)[/tex]
- [tex]\( f(200) = -4(200-50)(200-250) = -4(150)(-50) = 30000 \)[/tex]
- [tex]\( f(250) = -4(250-50)(250-250) = -4(200)(0) = 0 \)[/tex]
Values: [tex]\(-50000, 0, 30000, 40000, 30000, 0\)[/tex]
Clearly, these values don't match the given profit values.
2. [tex]\( f(x) = -\frac{1}{4}(x-50)(x-250) \)[/tex]
- [tex]\( f(0) = -\frac{1}{4}(0-50)(0-250) = -\frac{1}{4}(-50)(-250) = -3125 \)[/tex]
- [tex]\( f(50) = -\frac{1}{4}(50-50)(50-250) = -\frac{1}{4}(0)(-200) = 0 \)[/tex]
- [tex]\( f(100) = -\frac{1}{4}(100-50)(100-250) = -\frac{1}{4}(50)(-150) = 1875 \)[/tex]
- [tex]\( f(150) = -\frac{1}{4}(150-50)(150-250) = -\frac{1}{4}(100)(-100) = 2500 \)[/tex]
- [tex]\( f(200) = -\frac{1}{4}(200-50)(200-250) = -\frac{1}{4}(150)(-50) = 1875 \)[/tex]
- [tex]\( f(250) = -\frac{1}{4}(250-50)(250-250) = -\frac{1}{4}(200)(0) = 0 \)[/tex]
Values: [tex]\(-3125, 0, 1875, 2500, 1875, 0\)[/tex]
These values match exactly with the given profit values.
3. [tex]\( f(x) = 28(x+50)(x+250) \)[/tex]
- [tex]\( f(0) = 28(0+50)(0+250) = 28(50)(250) = 350000 \)[/tex]
- [tex]\( f(50) = 28(50+50)(50+250) = 28(100)(300) = 840000 \)[/tex]
- [tex]\( f(100) = 28(100+50)(100+250) = 28(150)(350) = 1470000 \)[/tex]
- [tex]\( f(150) = 28(150+50)(150+250) = 28(200)(400) = 2240000 \)[/tex]
- [tex]\( f(200) = 28(200+50)(200+250) = 28(250)(450) = 3150000 \)[/tex]
- [tex]\( f(250) = 28(250+50)(250+250) = 28(300)(500) = 4200000 \)[/tex]
These values do not match the given profit values.
4. [tex]\( f(x) = \frac{1}{28}(x+50)(x+250) \)[/tex]
- [tex]\( f(0) = \frac{1}{28}(0+50)(0+250) = \frac{1}{28}(50)(250) = 446.43 \)[/tex]
- [tex]\( f(50) = \frac{1}{28}(50+50)(50+250) = \frac{1}{28}(100)(300) = 1071.43 \)[/tex]
- [tex]\( f(100) = \frac{1}{28}(100+50)(100+250) = \frac{1}{28}(150)(350) = 1875 \)[/tex]
- [tex]\( f(150) = \frac{1}{28}(150+50)(150+250) = \frac{1}{28}(200)(400) = 2857.14 \)[/tex]
- [tex]\( f(200) = \frac{1}{28}(200+50)(200+250) = \frac{1}{28}(250)(450) = 4017.86 \)[/tex]
- [tex]\( f(250) = \frac{1}{28}(250+50)(250+250) = \frac{1}{28}(300)(500) = 5357.14 \)[/tex]
These values do not match the given profit values.
Upon evaluating, we see that the only function that matches the given profit values perfectly is:
[tex]\[ f(x) = -\frac{1}{4}(x-50)(x-250) \][/tex]
Hence, the function that can be used to model the monthly profit for [tex]\( x \)[/tex] trinkets produced is:
[tex]\[ \boxed{f(x) = -\frac{1}{4}(x-50)(x-250)} \][/tex]
The points given are:
[tex]\( (0, -3125) \)[/tex]
[tex]\( (50, 0) \)[/tex]
[tex]\( (100, 1875) \)[/tex]
[tex]\( (150, 2500) \)[/tex]
[tex]\( (200, 1875) \)[/tex]
[tex]\( (250, 0) \)[/tex]
We are provided with the following potential profit functions:
1. [tex]\( f(x) = -4(x-50)(x-250) \)[/tex]
2. [tex]\( f(x) = -\frac{1}{4}(x-50)(x-250) \)[/tex]
3. [tex]\( f(x) = 28(x+50)(x+250) \)[/tex]
4. [tex]\( f(x) = \frac{1}{28}(x+50)(x+250) \)[/tex]
We need to find out which one matches the provided points table.
For each function, we evaluate it at [tex]\( x = 0, 50, 100, 150, 200, 250 \)[/tex] to see if the function values [tex]\( y \)[/tex] match.
### Evaluating the Functions:
1. [tex]\( f(x) = -4(x-50)(x-250) \)[/tex]
- [tex]\( f(0) = -4(0-50)(0-250) = -4(-50)(-250) = -50000 \)[/tex]
- [tex]\( f(50) = -4(50-50)(50-250) = -4(0)(-200) = 0 \)[/tex]
- [tex]\( f(100) = -4(100-50)(100-250) = -4(50)(-150) = 30000 \)[/tex]
- [tex]\( f(150) = -4(150-50)(150-250) = -4(100)(-100) = 40000 \)[/tex]
- [tex]\( f(200) = -4(200-50)(200-250) = -4(150)(-50) = 30000 \)[/tex]
- [tex]\( f(250) = -4(250-50)(250-250) = -4(200)(0) = 0 \)[/tex]
Values: [tex]\(-50000, 0, 30000, 40000, 30000, 0\)[/tex]
Clearly, these values don't match the given profit values.
2. [tex]\( f(x) = -\frac{1}{4}(x-50)(x-250) \)[/tex]
- [tex]\( f(0) = -\frac{1}{4}(0-50)(0-250) = -\frac{1}{4}(-50)(-250) = -3125 \)[/tex]
- [tex]\( f(50) = -\frac{1}{4}(50-50)(50-250) = -\frac{1}{4}(0)(-200) = 0 \)[/tex]
- [tex]\( f(100) = -\frac{1}{4}(100-50)(100-250) = -\frac{1}{4}(50)(-150) = 1875 \)[/tex]
- [tex]\( f(150) = -\frac{1}{4}(150-50)(150-250) = -\frac{1}{4}(100)(-100) = 2500 \)[/tex]
- [tex]\( f(200) = -\frac{1}{4}(200-50)(200-250) = -\frac{1}{4}(150)(-50) = 1875 \)[/tex]
- [tex]\( f(250) = -\frac{1}{4}(250-50)(250-250) = -\frac{1}{4}(200)(0) = 0 \)[/tex]
Values: [tex]\(-3125, 0, 1875, 2500, 1875, 0\)[/tex]
These values match exactly with the given profit values.
3. [tex]\( f(x) = 28(x+50)(x+250) \)[/tex]
- [tex]\( f(0) = 28(0+50)(0+250) = 28(50)(250) = 350000 \)[/tex]
- [tex]\( f(50) = 28(50+50)(50+250) = 28(100)(300) = 840000 \)[/tex]
- [tex]\( f(100) = 28(100+50)(100+250) = 28(150)(350) = 1470000 \)[/tex]
- [tex]\( f(150) = 28(150+50)(150+250) = 28(200)(400) = 2240000 \)[/tex]
- [tex]\( f(200) = 28(200+50)(200+250) = 28(250)(450) = 3150000 \)[/tex]
- [tex]\( f(250) = 28(250+50)(250+250) = 28(300)(500) = 4200000 \)[/tex]
These values do not match the given profit values.
4. [tex]\( f(x) = \frac{1}{28}(x+50)(x+250) \)[/tex]
- [tex]\( f(0) = \frac{1}{28}(0+50)(0+250) = \frac{1}{28}(50)(250) = 446.43 \)[/tex]
- [tex]\( f(50) = \frac{1}{28}(50+50)(50+250) = \frac{1}{28}(100)(300) = 1071.43 \)[/tex]
- [tex]\( f(100) = \frac{1}{28}(100+50)(100+250) = \frac{1}{28}(150)(350) = 1875 \)[/tex]
- [tex]\( f(150) = \frac{1}{28}(150+50)(150+250) = \frac{1}{28}(200)(400) = 2857.14 \)[/tex]
- [tex]\( f(200) = \frac{1}{28}(200+50)(200+250) = \frac{1}{28}(250)(450) = 4017.86 \)[/tex]
- [tex]\( f(250) = \frac{1}{28}(250+50)(250+250) = \frac{1}{28}(300)(500) = 5357.14 \)[/tex]
These values do not match the given profit values.
Upon evaluating, we see that the only function that matches the given profit values perfectly is:
[tex]\[ f(x) = -\frac{1}{4}(x-50)(x-250) \][/tex]
Hence, the function that can be used to model the monthly profit for [tex]\( x \)[/tex] trinkets produced is:
[tex]\[ \boxed{f(x) = -\frac{1}{4}(x-50)(x-250)} \][/tex]
