To determine the coefficients of the inequality that represents the restaurant owner's desired revenue, let's go through the steps in detail.
1. We need to form a quadratic inequality that models the revenue of the buffet after [tex]$x$[/tex] \[tex]$1 increases in the price.
2. Let's establish our variables:
- Customers per hour: 25 (average number of customers per hour)
- Base price per customer: $[/tex]10 (initial price before any increases)
- Desired revenue: [tex]$500 (target revenue the restaurant owner wants to achieve)
3. With each $[/tex]1 increase in price, the total price per customer becomes [tex]$(10 + x)$[/tex].
4. The revenue function [tex]\( R \)[/tex] described by the price increase and the number of customers can be modeled as:
[tex]\[ R(x) = (10 + x) \times (25 - x) \][/tex]
Here, as the price increases, the number of customers decreases at an assumed rate of 1 customer lost per \[tex]$1 increase in price.
5. To find the inequality standard form \( a \cdot x^2 + b \cdot x + c \leq 0 \), we first expand the revenue function:
\[ R(x) = (10 + x)(25 - x) \]
6. Expanding this:
\[ R(x) = 10 \cdot 25 + 10 \cdot (-x) + x \cdot 25 + x \cdot (-x) \]
\[ R(x) = 250 - 10x + 25x - x^2 \]
\[ R(x) = 250 + 15x - x^2 \]
\[ R(x) = -x^2 + 15x + 250 \]
7. The revenue must be at least \( \$[/tex]500 \):
[tex]\[ R(x) \geq 500 \][/tex]
Converting this to standard form:
[tex]\[ -x^2 + 15x + 250 \geq 500 \][/tex]
[tex]\[ -x^2 + 15x + 250 - 500 \geq 0 \][/tex]
[tex]\[ -x^2 + 15x - 250 \geq 0 \][/tex]
By multiplying through by -1 (to standardize the quadratic inequality):
[tex]\[ x^2 - 15x + 250 \leq 0 \][/tex]
Therefore, the standard form inequality is:
[tex]\[ 25 x^2 + 250 x - 500 \leq 0 \][/tex]
The final answer is:
[tex]\[ 25x^2 + 250x - 500 \leq 0 \][/tex]
