Answer :
Let's solve the given problem step by step.
### Step 1: Identifying the Correct Inequality
We are given a few inequalities and need to determine which one describes the scenario of Sofia ordering an additional number of rolls, \( R \).
1. \( 12 + 24R \leq 100 \)
2. \( 12 + 24R \geq 100 \)
3. \( 24 + 2\pi R \leq 100 \)
4. \( 24 + 12R \geq 100 \)
Let's analyze them one by one:
- Inequality 1: \( 12 + 24R \leq 100 \)
This seems to represent a situation where the total cost of some combination of items \( (12 \text{ and } 24R) \) is less than or equal to 100.
- Inequality 2: \( 12 + 24R \geq 100 \)
This seems to represent a situation where the total cost of some combination of items \( (12 \text{ and } 24R) \) is greater than or equal to 100.
- Inequality 3: \( 24 + 2\pi R \leq 100 \)
This doesn't seem to fit the context of money spent on additional rolls, as it involves \( \pi \).
- Inequality 4: \( 24 + 12R \geq 100 \)
This seems to indicate another combination where the cost exceeds 100.
In our context, we need to identify the correct inequality that shows Sofia must spend at least a certain amount to get enough rolls. The correct inequality should thus depict a situation where the money spent must meet or exceed a certain value.
Answer: The correct inequality for this context is:
D) \( 24 + 12R \geq 100 \)
### Step 2: Solving the Inequality for the Least Cost
Now, let's solve the inequality \( 24 + 12R \geq 100 \) to find the minimum number of additional rolls \( R \) Sofia needs to order.
[tex]\[ 24 + 12R \geq 100 \][/tex]
First, subtract 24 from both sides:
[tex]\[ 12R \geq 100 - 24 \][/tex]
[tex]\[ 12R \geq 76 \][/tex]
Next, divide both sides by 12:
[tex]\[ R \geq \frac{76}{12} \][/tex]
[tex]\[ R \geq \frac{19}{3} \][/tex]
[tex]\[ R \geq 6.33\ldots \][/tex]
Since \( R \) represents the number of rolls, it must be an integer. Therefore, we round up to the next whole number, since Sofia can't order a fraction of a roll.
[tex]\[ R \geq 7 \][/tex]
### Step 3: Calculating the Least Amount of Additional Money Sofia Has to Spend
Given \( R = 7 \):
The cost per additional roll is [tex]$12$[/tex]. Therefore, the least amount of additional money Sofia can spend is:
[tex]\[ \text{Money spent} = 12 \times 7 = 84 \text{ dollars} \][/tex]
Answer: [tex]\( 84 \)[/tex] dollars.
### Step 1: Identifying the Correct Inequality
We are given a few inequalities and need to determine which one describes the scenario of Sofia ordering an additional number of rolls, \( R \).
1. \( 12 + 24R \leq 100 \)
2. \( 12 + 24R \geq 100 \)
3. \( 24 + 2\pi R \leq 100 \)
4. \( 24 + 12R \geq 100 \)
Let's analyze them one by one:
- Inequality 1: \( 12 + 24R \leq 100 \)
This seems to represent a situation where the total cost of some combination of items \( (12 \text{ and } 24R) \) is less than or equal to 100.
- Inequality 2: \( 12 + 24R \geq 100 \)
This seems to represent a situation where the total cost of some combination of items \( (12 \text{ and } 24R) \) is greater than or equal to 100.
- Inequality 3: \( 24 + 2\pi R \leq 100 \)
This doesn't seem to fit the context of money spent on additional rolls, as it involves \( \pi \).
- Inequality 4: \( 24 + 12R \geq 100 \)
This seems to indicate another combination where the cost exceeds 100.
In our context, we need to identify the correct inequality that shows Sofia must spend at least a certain amount to get enough rolls. The correct inequality should thus depict a situation where the money spent must meet or exceed a certain value.
Answer: The correct inequality for this context is:
D) \( 24 + 12R \geq 100 \)
### Step 2: Solving the Inequality for the Least Cost
Now, let's solve the inequality \( 24 + 12R \geq 100 \) to find the minimum number of additional rolls \( R \) Sofia needs to order.
[tex]\[ 24 + 12R \geq 100 \][/tex]
First, subtract 24 from both sides:
[tex]\[ 12R \geq 100 - 24 \][/tex]
[tex]\[ 12R \geq 76 \][/tex]
Next, divide both sides by 12:
[tex]\[ R \geq \frac{76}{12} \][/tex]
[tex]\[ R \geq \frac{19}{3} \][/tex]
[tex]\[ R \geq 6.33\ldots \][/tex]
Since \( R \) represents the number of rolls, it must be an integer. Therefore, we round up to the next whole number, since Sofia can't order a fraction of a roll.
[tex]\[ R \geq 7 \][/tex]
### Step 3: Calculating the Least Amount of Additional Money Sofia Has to Spend
Given \( R = 7 \):
The cost per additional roll is [tex]$12$[/tex]. Therefore, the least amount of additional money Sofia can spend is:
[tex]\[ \text{Money spent} = 12 \times 7 = 84 \text{ dollars} \][/tex]
Answer: [tex]\( 84 \)[/tex] dollars.
