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The table shows a company's profit based on the number of pounds of food produced.

Profit:
\begin{tabular}{|c|c|}
\hline Pounds of food produced & Profit (\[tex]$) \\
\hline 100 & $[/tex]-11,000[tex]$ \\
\hline 250 & 0 \\
\hline 500 & 10,300 \\
\hline 650 & 11,500 \\
\hline 800 & 9,075 \\
\hline
\end{tabular}

Using the quadratic regression model, which is the best estimate of the profit when 350 pounds of food are produced?

A. $[/tex]\[tex]$ 5,150$[/tex]
B. [tex]$\$[/tex] 5,300[tex]$
C. $[/tex]\[tex]$ 10,150$[/tex]
D. [tex]$\$[/tex] 11,000$


Sagot :

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].
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