To analyze the revenue generated by Blu-ray player sales based on the provided information, we can follow these steps:
-
Understanding the Revenue Function:
- Let \( x \) be the number of five-dollar decreases in price.
- The price of the Blu-ray player after \( x \) decreases is \( 100 - 5x \).
- The number of sales after \( x \) decreases is given by \( 32 + 2x \).
- Revenue \( R \) can be modeled with the equation: \[ R(x) = \text{(Price)} \times \text{(Sales)} = (100 - 5x)(32 + 2x) \]
- Expanding this formula: \[ R(x) = (100)(32) + (200 - 5x*32) - (5x)(2x) = 3200 + 200x - 160x - 10x^2 = -10x^2 + 40x + 3200 \]
- Thus, the revenue function is: \[ R(x) = -10x^2 + 40x + 3200 \]
-
Finding the Vertex:
- This function is a downward-opening parabola (as the coefficient of \( x^2 \) is negative).
- The vertex gives the maximum revenue and can be found using the formula for the \( x \)-coordinate of the vertex: \[ x = -\frac{b}{2a} \]
- Here, \( a = -10 \) and \( b = 40 \): \[ x = -\frac{40}{2(-10)} = \frac{40}{20} = 2 \]
- So, \( x = 2 \).
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Coordinates of the Vertex (Maximum Revenue):
- To find the maximum revenue, substitute \( x = 2 \) back into the revenue function: \[ R(2) = -10(2^2) + 40(2) + 3200 = -10(4) + 80 + 3200 \] \[ R(2) = -40 + 80 + 3200 = 3240 \]
- The vertex is at \( (2, 3240) \), thus:
- The coordinates of the vertex (maximum revenue) are \( (2, 3240) \).
-
Price for Maximum Revenue:
- The price at maximum revenue (when \( x = 2 \)): \[ P = 100 - 5(2) = 100 - 10 = 90 \]
- The price for maximum revenue is $90.
-
Determining No Revenue:
- Revenue will be zero when: \[ R(x) = 0 \]
- Solving for \( x \): \[ -10x^2 + 40x + 3200 = 0 \] \[ x^2 - 4x - 320 = 0 \]
- Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[
x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-320)}}{2(1)} = \frac{4 \pm \sqrt{16 + 1280}}{2} = \frac{4 \pm \sqrt{1296}}{2}
\]
\[
x = \frac{4 \pm 36}{2}
\]
- This gives: \[ x = \frac{40}{2} = 20 \quad \text{and} \quad x = \frac{-32}{2} = -16 \text{ (not valid)} \]
- Thus, the number of five-dollar decreases that would produce no revenue is \( x = 20 \).
Summary:
- Axis of Symmetry: \( x = 2 \)
- Coordinates of Vertex (Max Revenue): \( (2, 3240) \)
- Price for Maximum Revenue: $90
- Number of Five Dollar Decreases that Produce No Revenue: 20
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