Let's start by analyzing the functions you've provided:
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Total Revenue (R): \[ R = 870x \] This is a linear function because it has the form \(y = mx + b\), where \(m\) is the slope (in this case, 870) and \(b\) is the y-intercept (0 here).
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Total Cost (C): \[ C = 30,000 + 50x + x^2 \] This is a quadratic function because it contains a term with \(x^2\).
Part 1
A. Which function is quadratic, and which is linear?
The correct answer is: A. The total cost function is quadratic and the total revenue function is linear.
Part 2
B. Find the profit function for this product.
The profit function (P) is defined as the difference between total revenue (R) and total cost (C): \[ P = R - C \] Substituting the expressions for R and C: \[ P = 870x - (30,000 + 50x + x^2) \] Now simplify the equation: \[ P = 870x - 30,000 - 50x - x^2 \] \[ P = (870x - 50x) - x^2 - 30,000 \] \[ P = 820x - x^2 - 30,000 \] Thus, the profit function is: \[ P = -x^2 + 820x - 30,000 \]
Part 3
C. Is the profit function a linear function, a quadratic function, or neither?
We can observe that the profit function \(P = -x^2 + 820x - 30,000\) contains a term with \(x^2\), which makes it a quadratic function.
Summary
- Part 1: Total cost is quadratic; total revenue is linear.
- Part 2: The profit function \(P = -x^2 + 820x - 30,000\).
- Part 3: The profit function is a quadratic function.