To determine the number of burgers a restaurant expects to sell this year, we need to analyze the given information and the options presented.
1. Initial Sales and Growth Rate:
- The restaurant sold 9,000 burgers in January.
- The sales are expected to increase by [tex]\(4.8\%\)[/tex] per month. This [tex]\(4.8\%\)[/tex] growth per month translates to a monthly growth multiplier of [tex]\(1.048\)[/tex].
2. Interpreting the Growth Rate:
- The initial sales for January would be [tex]\(9000\)[/tex].
- For February, the sales would be [tex]\(9000 \times 1.048\)[/tex].
- For March, the sales would be [tex]\(9000 \times (1.048)^2\)[/tex], and so on.
3. Summing the Sales Over 12 Months:
- We will compute the cumulative sales from January to December. This involves summing the sales where each month’s sales are growing exponentially due to the [tex]\(4.8\%\)[/tex] growth rate.
Given the growth rate and initial sales, we use a geometric series to calculate the total sales over 12 months. The correct summation formula to use is:
[tex]\[ \sum_{n=0}^{11} (1.048)^n \times 9000 \][/tex]
Let's evaluate the other choices to confirm why they are incorrect:
- [tex]\(\sum_{n=1}^{12} (1.048)^n \times 9000\)[/tex]:
This formula starts the summation from [tex]\( n = 1 \)[/tex] instead of [tex]\( n = 0 \)[/tex]. Since we need to include the sales from January, this option does not properly account for the sales starting from January.
- [tex]\(\sum_{n=1}^{12} (4.8n + 9000)\)[/tex]:
This formula suggests a linear growth model, using a constant increment rather than percentage growth, which does not align with the exponential growth described.
- [tex]\(\sum_{n=0}^{12} 4.8n + 9000\)[/tex]:
This formula also represents a linear growth model and incorrectly sums an arithmetic sequence instead of an exponential one.
Therefore, the correct formula to use for calculating the total number of burgers the restaurant expects to sell this year is:
[tex]\[ \sum_{n=0}^{11} (1.048)^n \times 9000 \][/tex]
Now, summing this series gives us the total expected sales for the year. Given that the result of the calculations for this formula is:
[tex]\[ \sum_{n=0}^{11} (1.048)^n \times 9000 = 141,606.65455068968 \][/tex]
Thus, the correct expected sales total for the year is approximately:
[tex]\[ 141,606.65 \text{ burgers} \][/tex]
And finally, the correct formula among the given options is:
[tex]\[ \boxed{\sum_{n=0}^{11} (1.048)^n \times 9000} \][/tex]
