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Zorluk: OrtaQuadratic Functions and Graphs

A manufacturer models the daily profit, P(x)P(x), in dollars, from producing and selling xx units of a product using the function P(x)=−2x2+kx−800P(x) = -2x^2 + kx - 800, where kk is a constant. If the maximum daily profit is 1,0001,000 dollars, what is the number of units that must be sold to achieve this maximum profit?

  1. A
    10
  2. 30Cevap
  3. C
    60
  4. D
    120

Cevap

The manufacturer must sell 30 units to achieve the maximum daily profit.
The correct answer is 30. The maximum value of a downward-opening quadratic function occurs at its vertex (h,q)(h, q). Given that the maximum value is 1000, we write the function in vertex form: P(x)=−2(x−h)2+1000P(x) = -2(x - h)^2 + 1000. Expanding this gives P(x)=−2x2+4hx−2h2+1000P(x) = -2x^2 + 4hx - 2h^2 + 1000. Comparing this to the given function P(x)=−2x2+kx−800P(x) = -2x^2 + kx - 800, the constant term must satisfy −2h2+1000=−800-2h^2 + 1000 = -800. Solving for hh yields 2h2=18002h^2 = 1800, so h2=900h^2 = 900. Taking the positive square root because the number of units must be positive gives h=30h = 30.

Adım Adım Çözüm

1
Write the quadratic function in vertex form and identify the given parameters.
The vertex form of a quadratic function is P(x)=a(x−h)2+qP(x) = a(x-h)^2 + q, where (h,q)(h, q) is the vertex. Here, a=−2a = -2 and the maximum profit q=1000q = 1000, so the function is P(x)=−2(x−h)2+1000P(x) = -2(x-h)^2 + 1000.
Since the coefficient of the x2x^2 term is negative (a=−2a = -2), the parabola opens downward, meaning the vertex represents the maximum value of the function.
2
Expand the vertex form equation to compare its coefficients with the standard form P(x)=−2x2+kx−800P(x) = -2x^2 + kx - 800.
P(x)=−2(x2−2hx+h2)+1000=−2x2+4hx−(2h2−1000)P(x) = -2(x^2 - 2hx + h^2) + 1000 = -2x^2 + 4hx - (2h^2 - 1000)
Expanding the vertex form allows us to equate corresponding coefficients and constant terms between the two forms of the quadratic function.
3
Equate the constant terms from both expressions to solve for the vertex xx-coordinate hh.
−2h2+1000=−800⇒−2h2=−1800⇒h2=900⇒h=30-2h^2 + 1000 = -800 \Rightarrow -2h^2 = -1800 \Rightarrow h^2 = 900 \Rightarrow h = 30 (since the number of units must be positive).
By setting the constant terms equal, we can isolate and solve for hh, which represents the number of units that maximizes the daily profit.

Anahtar Kavram

Quadratic functions in vertex form and their standard form equivalents.
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Konular

Quadratic Functions and Graphs

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