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Difficulty: MediumQuadratic Functions and Graphs

A company's daily profit, P(x)P(x), in dollars, from selling xx units of a product is modeled by the function P(x)=−2x2+kx−800P(x) = -2x^2 + kx - 800, where kk is a constant. If the company achieves its maximum daily profit of $1000\$1000 when it sells 3030 units, what is the value of kk?

  1. 120Answer
  2. B
    60
  3. C
    -120
  4. D
    180

Answer

120
The maximum value of a quadratic function occurs at its vertex. Given that the vertex is at (30,1000)(30, 1000) and the coefficient of the squared term is −2-2, the profit function can be written in vertex form as P(x)=−2(x−30)2+1000P(x) = -2(x - 30)^2 + 1000. Expanding this expression yields P(x)=−2(x2−60x+900)+1000=−2x2+120x−800P(x) = -2(x^2 - 60x + 900) + 1000 = -2x^2 + 120x - 800. Comparing this to the given equation P(x)=−2x2+kx−800P(x) = -2x^2 + kx - 800, the coefficient of xx must be 120. Alternatively, using the vertex formula h=−b/(2a)h = -b/(2a) with h=30h = 30 and a=−2a = -2 gives 30=−k/(2×−2)30 = -k/(2 \times -2), which simplifies to 30=k/430 = k/4 and results in k=120k = 120.

Step-by-Step Solution

1
Identify the vertex coordinates from the given context.
The vertex of the profit parabola is at (h,d)=(30,1000)(h, d) = (30, 1000).
The maximum profit of $1000\$1000 occurs when 3030 units are sold, representing the peak of the downward-opening parabola.
2
Substitute the vertex and the leading coefficient a=−2a = -2 into the vertex form of a quadratic function, P(x)=a(x−h)2+dP(x) = a(x - h)^2 + d.
The equation becomes P(x)=−2(x−30)2+1000P(x) = -2(x - 30)^2 + 1000.
The vertex form allows direct substitution of the vertex coordinates to build the function's equation.
3
Expand the vertex form equation into standard form.
P(x)=−2(x2−60x+900)+1000=−2x2+120x−800P(x) = -2(x^2 - 60x + 900) + 1000 = -2x^2 + 120x - 800.
Expanding the equation allows direct comparison of terms with the given standard form P(x)=−2x2+kx−800P(x) = -2x^2 + kx - 800.
4
Compare the coefficient of the xx term in the expanded equation to the coefficient of the xx term in the given equation.
k=120k = 120.
Corresponding coefficients of identical functions must be equal, allowing us to determine the value of the constant kk.

Key Concept

Quadratic Functions and Graphs
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Topics

Quadratic Functions and Graphs

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