A pottery store owner determines that the revenue for sales of a particular item can be modeled by the function `r(x) = 50 sqrt(x) - 40`, where x is the number of the items sold. How many of the item must be sold to generate $110 in revenue?

Options

A. 5
B. 6
C. 7
D. 9

Answer

To find out how many items must be sold to generate $110 in revenue, we start with the revenue function given by:

\[ r(x) = 50 \sqrt{x} - 40 \]

where \( r(x) \) is the revenue and \( x \) is the number of items sold. We want to find \( x \) when the revenue \( r(x) \) is $110.

### Step-by-Step Solution:

1. **Set up the equation**:
We know that the revenue is $110, so we set the function equal to 110:
\[ 110 = 50 \sqrt{x} - 40 \]

2. **Add 40 to both sides**:
To isolate the term with the square root, we add 40 to both sides of the equation:
\[ 110 + 40 = 50 \sqrt{x} \]
\[ 150 = 50 \sqrt{x} \]

3. **Divide both sides by 50**:
Next, we divide both sides by 50 to solve for \( \sqrt{x} \):
\[ \frac{150}{50} = \sqrt{x} \]
\[ 3 = \sqrt{x} \]

4. **Square both sides**:
To eliminate the square root, we square both sides of the equation:
\[ 3^2 = x \]
\[ 9 = x \]

5. **Conclusion**:
Therefore, the number of items that must be sold to generate $110 in revenue is:
\[ x = 9 \]

### Final Answer:
The pottery store owner must sell **9 items** to generate $110 in revenue.

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