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Difficulty: MediumQuadratic Equations and the Quadratic Formula

A model glider is launched from a hill. Its height h(t)h(t), in meters above the valley floor tt seconds after launch, is modeled by the function h(t)=−0.5t2+3.5t+10h(t) = -0.5t^2 + 3.5t + 10. Which of the following is a possible value of tt, in seconds, when the glider is at a height of exactly 1212 meters?

  1. 7+332\frac{7 + \sqrt{33}}{2}Answer
  2. B
    7+652\frac{7 + \sqrt{65}}{2}
  3. C
    −7+332\frac{-7 + \sqrt{33}}{2}
  4. D
    7+337 + \sqrt{33}
  5. E
    1111

Answer

7+332\frac{7 + \sqrt{33}}{2}
The correct answer is obtained by setting the height equation h(t)=12h(t) = 12, which simplifies to −0.5t2+3.5t−2=0-0.5t^2 + 3.5t - 2 = 0. Multiplying by −2-2 gives the standard form t2−7t+4=0t^2 - 7t + 4 = 0. Applying the quadratic formula with a=1a = 1, b=−7b = -7, and c=4c = 4 gives t=7±332t = \frac{7 \pm \sqrt{33}}{2}. Thus, the option representing 7+332\frac{7 + \sqrt{33}}{2} is the correct choice.

Step-by-Step Solution

1
Set the height function equal to the target height of 12 meters.
−0.5t2+3.5t+10=12-0.5t^2 + 3.5t + 10 = 12
To find when the glider reaches exactly 12 meters, we set the model function equal to 12.
2
Subtract 12 from both sides to set the quadratic equation to zero.
−0.5t2+3.5t−2=0-0.5t^2 + 3.5t - 2 = 0
A quadratic equation must be in standard form at2+bt+c=0at^2 + bt + c = 0 before applying the quadratic formula.
3
Multiply the entire equation by −2-2 to eliminate decimal coefficients.
t2−7t+4=0t^2 - 7t + 4 = 0
Working with integer coefficients reduces calculation errors when applying the quadratic formula. Here, a=1a = 1, b=−7b = -7, and c=4c = 4.
4
Apply the quadratic formula t=−b±b2−4ac2at = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
t=−(−7)±(−7)2−4(1)(4)2(1)=7±49−162=7±332t = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(1)(4)}}{2(1)} = \frac{7 \pm \sqrt{49 - 16}}{2} = \frac{7 \pm \sqrt{33}}{2}
The quadratic formula is used to solve quadratic equations that cannot be easily factored using integers.

Key Concept

Solving quadratic equations with decimal coefficients by converting to standard integer form and applying the quadratic formula.
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Topics

Quadratic Equations and the Quadratic Formula

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