To solve this problem and find the best estimate of the profit when 350 pounds of food are produced, we'll fit a quadratic regression model to the given data points and then use this model to estimate the profit.
Here are the steps:
1. Data Points: We have the following data points:
- For 100 pounds of food produced, the profit is -[tex]$11,000.
- For 250 pounds of food produced, the profit is $[/tex]0.
- For 500 pounds of food produced, the profit is [tex]$10,300.
- For 650 pounds of food produced, the profit is $[/tex]11,500.
- For 800 pounds of food produced, the profit is [tex]$9,075.
2. Quadratic Model: We assume that the profit \( y \) can be modeled by a quadratic function of the form:
\[ y = ax^2 + bx + c \]
where \( x \) is the number of pounds of food produced, and \( a \), \( b \), and \( c \) are coefficients to be determined.
3. Determining Coefficients: Using regression techniques, the coefficients for the quadratic model are found to be:
\[ a = -0.08170535818970577 \]
\[ b = 102.2355199906203 \]
\[ c = -20420.95497713684 \]
4. Estimating Profit for 350 Pounds: We now calculate the profit when 350 pounds of food are produced. Plugging \( x = 350 \) into the quadratic model, we get:
\[ y = a(350)^2 + b(350) + c \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ y = -0.08170535818970577 \cdot (350)^2 + 102.2355199906203 \cdot 350 - 20420.95497713684 \]
5. Result: The computed profit is approximately:
\[ y \approx 5352.57 \]
Based on this calculation, the best estimate of the profit when 350 pounds of food are produced is approximately \$[/tex]5,352.57.
Given the provided answer choices:
- \[tex]$5,150
- \$[/tex]5,300
- \[tex]$10,150
- \$[/tex]11,000
The closest match to our estimated profit of \[tex]$5,352.57 is:
\[ \$[/tex]5,300 \]
Therefore, the best estimate of the profit when 350 pounds of food are produced is [tex]\(\boxed{5,300}\)[/tex].
