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Answer :
the professor should plan on ordering 24 books for the class of 25 students.
To determine how many books the professor should order, we need to find the number of books demanded by the students and then account for potential excess books. Here's the calculation:
1. Calculate the expected number of students who will buy books:
[tex]\[ \text{Expected number of students} = 25 \times \text{Percentage of students who bought a book} \][/tex]
2. Calculate the expected number of books demanded:
[tex]\[ \text{Expected demand} = \text{Expected number of students} + \text{Excess demand} \][/tex]
3. Calculate the excess demand:
[tex]\[ \text{Excess demand} = (\text{Expected number of students} - 25) \times \text{Percentage of excess demand} \][/tex]
4. Calculate the total cost of ordering books:
[tex]\[ \text{Total cost} = (\text{Number of books ordered} \times \text{Cost to print}) +[/tex][tex](\text{Shipping cost} \times \text{Number of excess books})[/tex]
5. Determine the optimal number of books to order:
[tex]\[ \text{Number of books to order} = \text{Expected demand} + \text{Number of excess books} \][/tex]
Now, let's perform the calculations:
- Expected number of students who will buy books:
[tex]\[ 25 \times (0.04 + 0.15 + 0.17 + 0.18 + 0.26 + 0.1 + 0.06 + 0.04) \][/tex]
[tex]\[ = 25 \times 0.96 = 24 \][/tex]
- Expected demand:
[tex]\[ \text{Expected demand} = 24 + \text{Excess demand} \][/tex]
- Excess demand:
[tex]\[ \text{Excess demand} = (24 - 25) \times (0.04 + 0.15 + 0.17 + 0.18 + 0.26 + 0.1 + 0.06 + 0.04) \][/tex]
[tex]\[ = -1 \times (0.04 + 0.15 + 0.17 + 0.18 + 0.26 + 0.1 + 0.06 + 0.04) = -1 \times 0.96 = -0.96 \][/tex]
Since the excess demand is negative, it means there will be fewer excess books.
- Total cost of ordering books:
[tex]\[ \text{Total cost} = (\text{Number of books ordered} \times \text{Cost to print})[/tex] [tex]+ (\text{Shipping cost} \times \text{Number of excess books})[/tex]
Given that each custom book costs $12 to print and the shipping cost is $1 per book, we can calculate the total cost for different scenarios and determine the optimal number of books to order. Let's calculate it.
Let's calculate the total cost for different scenarios and determine the optimal number of books to order:
1. For 24 books ordered:
[tex]\[ \text{Total cost} = (24 \times 12) + (0 \times 1) = 288 \][/tex]
2. For 25 books ordered:
[tex]\[ \text{Total cost} = (25 \times 12) + (1 \times 1) = 301 \][/tex]
3. For 26 books ordered:
[tex]\[ \text{Total cost} = (26 \times 12) + (1 \times 1) = 313 \][/tex]
Since we want to minimize costs, we choose the option with the lowest total cost. In this case, ordering 24 books yields the lowest total cost. Therefore, the professor should plan on ordering 24 books for the class of 25 students.
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