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To determine the cost [tex]\( C(x) \)[/tex] of purchasing [tex]\( x \)[/tex] t-shirts for a school fair, we need to consider the pricing structure provided by the supplier. Here's a detailed step-by-step explanation of how the piecewise function works and what the costs would be for specific values of [tex]\( x \)[/tex]:
1. For [tex]\( x \leq 0 \)[/tex]: If [tex]\( x \)[/tex] is less than or equal to 0, no t-shirts are being purchased, and in a practical sense, the cost should be undefined because you can't buy a negative number of t-shirts. The function should return a value indicating the purchase is invalid:
[tex]\[ C(x) = \text{None} \quad \text{for} \ x \leq 0 \][/tex]
2. For [tex]\( 0 < x \leq 100 \)[/tex]: If [tex]\( x \)[/tex] is between 1 and 100 (inclusive), the cost per t-shirt is P165.00. The total cost [tex]\( C(x) \)[/tex] in this case can be calculated as:
[tex]\[ C(x) = 165x \quad \text{for} \ 1 \leq x \leq 100 \][/tex]
3. For [tex]\( x = 100 \)[/tex]: If exactly 100 t-shirts are being purchased, there is a bulk deal where the cost is fixed at P13,000.00:
[tex]\[ C(x) = 13000 \quad \text{for} \ x = 100 \][/tex]
4. For [tex]\( x > 100 \)[/tex]: For any number of t-shirts greater than 100, the first 100 t-shirts are covered by the bulk price of P13,000.00, and each additional t-shirt costs P115.00. The total cost for [tex]\( x \)[/tex] t-shirts where [tex]\( x \)[/tex] is more than 100 can be calculated using:
[tex]\[ C(x) = 13000 + 115(x - 100) \quad \text{for} \ x > 100 \][/tex]
Now let's compile this into the piecewise function:
[tex]\[ C(x) = \begin{cases} \text{None} & x \leq 0 \\ 165x & 0 < x < 100 \\ 13000 & x = 100 \\ 13000 + 115(x - 100) & x > 100 \end{cases} \][/tex]
### Example Calculations
1. For [tex]\( x = 50 \)[/tex]:
- The number of t-shirts ordered is 50.
- Since [tex]\( 0 < x < 100 \)[/tex], we use the formula [tex]\( C(x) = 165x \)[/tex]:
[tex]\[ C(50) = 165 \times 50 = 8250 \][/tex]
2. For [tex]\( x = 100 \)[/tex]:
- The number of t-shirts ordered is 100.
- Since [tex]\( x = 100 \)[/tex], the total cost is the bulk price:
[tex]\[ C(100) = 13000 \][/tex]
3. For [tex]\( x = 150 \)[/tex]:
- The number of t-shirts ordered is 150.
- Since [tex]\( x > 100 \)[/tex], we use the formula [tex]\( C(x) = 13000 + 115(x - 100) \)[/tex]:
[tex]\[ C(150) = 13000 + 115(150 - 100) = 13000 + 115 \times 50 = 13000 + 5750 = 18750 \][/tex]
Therefore, the costs for 50, 100, and 150 t-shirts are 8250, 13000, and 18750 respectively.
1. For [tex]\( x \leq 0 \)[/tex]: If [tex]\( x \)[/tex] is less than or equal to 0, no t-shirts are being purchased, and in a practical sense, the cost should be undefined because you can't buy a negative number of t-shirts. The function should return a value indicating the purchase is invalid:
[tex]\[ C(x) = \text{None} \quad \text{for} \ x \leq 0 \][/tex]
2. For [tex]\( 0 < x \leq 100 \)[/tex]: If [tex]\( x \)[/tex] is between 1 and 100 (inclusive), the cost per t-shirt is P165.00. The total cost [tex]\( C(x) \)[/tex] in this case can be calculated as:
[tex]\[ C(x) = 165x \quad \text{for} \ 1 \leq x \leq 100 \][/tex]
3. For [tex]\( x = 100 \)[/tex]: If exactly 100 t-shirts are being purchased, there is a bulk deal where the cost is fixed at P13,000.00:
[tex]\[ C(x) = 13000 \quad \text{for} \ x = 100 \][/tex]
4. For [tex]\( x > 100 \)[/tex]: For any number of t-shirts greater than 100, the first 100 t-shirts are covered by the bulk price of P13,000.00, and each additional t-shirt costs P115.00. The total cost for [tex]\( x \)[/tex] t-shirts where [tex]\( x \)[/tex] is more than 100 can be calculated using:
[tex]\[ C(x) = 13000 + 115(x - 100) \quad \text{for} \ x > 100 \][/tex]
Now let's compile this into the piecewise function:
[tex]\[ C(x) = \begin{cases} \text{None} & x \leq 0 \\ 165x & 0 < x < 100 \\ 13000 & x = 100 \\ 13000 + 115(x - 100) & x > 100 \end{cases} \][/tex]
### Example Calculations
1. For [tex]\( x = 50 \)[/tex]:
- The number of t-shirts ordered is 50.
- Since [tex]\( 0 < x < 100 \)[/tex], we use the formula [tex]\( C(x) = 165x \)[/tex]:
[tex]\[ C(50) = 165 \times 50 = 8250 \][/tex]
2. For [tex]\( x = 100 \)[/tex]:
- The number of t-shirts ordered is 100.
- Since [tex]\( x = 100 \)[/tex], the total cost is the bulk price:
[tex]\[ C(100) = 13000 \][/tex]
3. For [tex]\( x = 150 \)[/tex]:
- The number of t-shirts ordered is 150.
- Since [tex]\( x > 100 \)[/tex], we use the formula [tex]\( C(x) = 13000 + 115(x - 100) \)[/tex]:
[tex]\[ C(150) = 13000 + 115(150 - 100) = 13000 + 115 \times 50 = 13000 + 5750 = 18750 \][/tex]
Therefore, the costs for 50, 100, and 150 t-shirts are 8250, 13000, and 18750 respectively.
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