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The cost, [tex]\$C[/tex], of producing [tex]x[/tex] units of a product is given by the function [tex]C = 6000 + 2x[/tex], up to a cost of [tex]\$10,000[/tex].

(a) Find the domain.
The domain is [tex]\square \leq x \leq \square[/tex]

(b) Find the range.
The range is [tex]\square \leq C \leq \square[/tex]

Sagot :

GinnyAnswer
Certainly! Let's go through this step-by-step.

### Part (a) Finding the Domain

The cost function is given as [tex]\( C = 6000 + 2x \)[/tex]. This function calculates the cost [tex]\( C \)[/tex] for producing [tex]\( x \)[/tex] units of a product.

1. Identify the given cost limits:
- The fixed initial cost is [tex]$6000. - The variable cost per unit produced is $[/tex]2.
- The maximum cost is $10000.

2. Determine the domain of [tex]\( x \)[/tex]:
- The domain represents the possible values for [tex]\( x \)[/tex] (the number of units produced).
- Start with the minimum value for [tex]\( x \)[/tex]:
[tex]\[ x_{\text{min}} = 0 \][/tex]
This is because it is usually logical to assume that producing zero units is possible.

- Now, find the maximum value for [tex]\( x \)[/tex]:
- Set the cost function to the maximum cost:
[tex]\[ 6000 + 2x = 10000 \][/tex]
- Solve for [tex]\( x \)[/tex]:
[tex]\[ 2x = 10000 - 6000 \][/tex]
[tex]\[ 2x = 4000 \][/tex]
[tex]\[ x = \frac{4000}{2} = 2000 \][/tex]
- So, the maximum value for [tex]\( x \)[/tex] is 2000.

3. Express the domain:
- Combining these results, the domain is:
[tex]\[ 0 \leq x \leq 2000 \][/tex]

### Part (b) Finding the Range

The range of a function represents the possible values that the output (in this case, [tex]\( C \)[/tex]) can take.

1. Determine the minimum value of [tex]\( C \)[/tex]:
- The minimum cost occurs when [tex]\( x = 0 \)[/tex]:
[tex]\[ C_{\text{min}} = 6000 + 2 \cdot 0 = 6000 \][/tex]

2. Determine the maximum value of [tex]\( C \)[/tex]:
- The maximum cost corresponds to the maximum [tex]\( x \)[/tex]:
[tex]\[ x = 2000 \][/tex]
- Using the cost function:
[tex]\[ C_{\text{max}} = 6000 + 2 \cdot 2000 = 6000 + 4000 = 10000 \][/tex]

3. Express the range:
- Combining these results, the range is:
[tex]\[ 6000 \leq C \leq 10000 \][/tex]

### Final Answer

Putting all this information together:

(a) The domain is:
[tex]\[ 0 \leq x \leq 2000 \][/tex]

(b) The range is:
[tex]\[ 6000 \leq C \leq 10000 \][/tex]

This concludes the detailed step-by-step solution to the problem.
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