To estimate the profit when 350 pounds of food are produced using a quadratic regression model, we follow these steps:
1. Collect the Data:
Given:
- Pounds of food produced: [tex]\[100, 250, 500, 650, 800\][/tex]
- Corresponding profits: [tex]\[-11,000, 0, 10,300, 11,500, 9,075\][/tex]
2. Fit a Quadratic Model:
A quadratic model can be expressed as:
[tex]\[
y = ax^2 + bx + c
\][/tex]
where [tex]\( x \)[/tex] represents the pounds of food and [tex]\( y \)[/tex] represents the profit.
3. Determine Coefficients:
By fitting a quadratic model to the given data, we find the coefficients of the quadratic equation. The coefficients are:
[tex]\[
a = -0.0817, \quad b = 102.2355, \quad c = -20420.955
\][/tex]
4. Formulate the Quadratic Equation:
Substituting the coefficients into the quadratic equation gives:
[tex]\[
y = -0.0817x^2 + 102.2355x - 20420.955
\][/tex]
5. Estimate the Profit for 350 pounds:
To find the profit for 350 pounds of food produced, we substitute [tex]\( x = 350 \)[/tex] into the quadratic equation:
[tex]\[
y = -0.0817(350)^2 + 102.2355(350) - 20420.955
\][/tex]
6. Calculate the Estimated Profit:
Performing the calculations (and using the precise result obtained from the previous analysis):
[tex]\[
y \approx 5352.57
\][/tex]
Therefore, the estimated profit when 350 pounds of food are produced is approximately \$5,352.57. Among the given options, the closest estimate is:
[tex]\[
\boxed{5,300}
\][/tex]
