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Answer :
To determine the cost of purchasing safety goggles for more than 150 pairs, we need to establish a linear function that considers the initial cost for the first 150 pairs and the additional cost for each pair beyond that point.
Here's how we can develop the function step-by-step:
1. Understand the Initial Cost:
- The first 150 pairs of safety goggles have a fixed cost of [tex]$337.50.
2. Calculate Additional Costs:
- Any pair beyond those initial 150 pairs costs $[/tex]2.10 each.
3. Define the Linear Function:
- For any number of pairs, [tex]\( x \)[/tex], where [tex]\( x \)[/tex] is greater than 150, the total cost can be calculated starting with the initial cost, and then adding the cost for the additional goggles.
- The number of additional pairs (beyond the initial 150) is [tex]\( x - 150 \)[/tex].
- Therefore, the cost of these additional pairs is [tex]\( (x - 150) \times 2.10 \)[/tex].
4. Construct the Function:
- The total cost, [tex]\( f(x) \)[/tex], for [tex]\( x \)[/tex] pairs of goggles is the sum of the cost for the first 150 pairs and the cost of the additional pairs.
- So the function is:
[tex]\[
f(x) = 337.50 + (x - 150) \times 2.10
\][/tex]
This function will give you the total cost for any number of pairs [tex]\( x \)[/tex] that is greater than 150.
Here's how we can develop the function step-by-step:
1. Understand the Initial Cost:
- The first 150 pairs of safety goggles have a fixed cost of [tex]$337.50.
2. Calculate Additional Costs:
- Any pair beyond those initial 150 pairs costs $[/tex]2.10 each.
3. Define the Linear Function:
- For any number of pairs, [tex]\( x \)[/tex], where [tex]\( x \)[/tex] is greater than 150, the total cost can be calculated starting with the initial cost, and then adding the cost for the additional goggles.
- The number of additional pairs (beyond the initial 150) is [tex]\( x - 150 \)[/tex].
- Therefore, the cost of these additional pairs is [tex]\( (x - 150) \times 2.10 \)[/tex].
4. Construct the Function:
- The total cost, [tex]\( f(x) \)[/tex], for [tex]\( x \)[/tex] pairs of goggles is the sum of the cost for the first 150 pairs and the cost of the additional pairs.
- So the function is:
[tex]\[
f(x) = 337.50 + (x - 150) \times 2.10
\][/tex]
This function will give you the total cost for any number of pairs [tex]\( x \)[/tex] that is greater than 150.
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