To determine the number of burgers the restaurant expects to sell this year, let's break down the requirements and details given in the problem:
1. Initial Sales in January: The restaurant sells 9,000 burgers in January.
2. Monthly Growth Rate: Sales are expected to grow at a rate of 4.8% per month.
3. Total Duration: The growth will be calculated over the next 12 months.
Given these pieces of information, we need to determine which formula properly calculates the total number of burgers sold over the 12 months.
### Step-by-Step Solution:
#### 1. Understanding Growth Rate Formula
The sales in consecutive months are a geometric progression where each month's sales is 4.8% more than the previous month's sales. The monthly growth factor can be expressed as:
[tex]\[ 1.048 = 1 + 0.048 \][/tex]
#### 2. Monthly Sales Calculation
Let's denote the number of burgers sold each month as follows:
- January: [tex]\( 9,000 \)[/tex]
- February: [tex]\( 9,000 \times 1.048 \)[/tex]
- March: [tex]\( 9,000 \times (1.048)^2 \)[/tex]
- …
- December: [tex]\( 9,000 \times (1.048)^{11} \)[/tex]
Each subsequent month’s sales are multiplied by the growth factor raised to the power of the month (starting from 0 for January). Therefore, the number of burgers sold in month [tex]\( n \)[/tex] can be described generally as:
[tex]\[ \text{Burgers Sold in Month } n = 9,000 \times (1.048)^n \][/tex]
#### 3. Total Burgers Sold Over the Year
To find the total number of burgers sold over the entire year, we sum the burgers sold each month. This sum is given by:
[tex]\[ \sum_{n=0}^{11} 9,000 \times (1.048)^n \][/tex]
#### 4. Evaluate the Provided Formulas
Now let’s compare the given formulas to our derived formula:
- Option 1: [tex]\(\sum_{n=0}^{11}(1.048)^n(9,000)\)[/tex]
- This matches our derived formula exactly.
- Option 2: [tex]\(\sum_{n=1}^{12}(1.048)^n(9,000)\)[/tex]
- This does not match because it starts at [tex]\( n = 1 \)[/tex] and goes to [tex]\( n = 12 \)[/tex] which is off by one step in the growth process.
- Option 3: [tex]\(\sum_{n=1}^{12}(4.8 n + 9,000)\)[/tex]
- This represents an arithmetic sequence, not a geometric progression, so it is incorrect.
- Option 4: [tex]\(\sum_{n=0}^{12} 4.8 n + 9,000\)[/tex]
- This is an arithmetic sequence and does not properly account for the compounding growth rate, so it is incorrect.
### Answer:
The correct formula to determine the number of burgers the restaurant expects to sell this year is:
[tex]\[ \sum_{n=0}^{11}(1.048)^n(9,000) \][/tex]
