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Difficulty: HardQuadratic Equations and the Quadratic Formula

A model rocket is launched vertically upward from a platform. Its height, hh, in meters, tt seconds after launch is modeled by the equation h=−4.9t2+v0t+2h = -4.9t^2 + v_0 t + 2, where v0v_0 is the constant initial upward velocity in meters per second. If the rocket never reaches a height of 1212 meters, which of the following inequalities represents all possible values of v0v_0?

  1. A
    0<v0<70 < v_0 < 7
  2. B
    0<v0<9.80 < v_0 < 9.8
  3. 0<v0<140 < v_0 < 14Answer
  4. D
    0<v0<19.60 < v_0 < 19.6
  5. E
    0<v0<280 < v_0 < 28

Answer

0<v0<140 < v_0 < 14
The correct answer is the range of values where the velocity is between 0 and 14. Setting the rocket's height equal to 12 meters gives the equation −4.9t2+v0t+2=12-4.9t^2 + v_0 t + 2 = 12. Subtracting 12 from both sides results in −4.9t2+v0t−10=0-4.9t^2 + v_0 t - 10 = 0. Since the rocket never reaches 12 meters, this quadratic equation has no real solutions, meaning its discriminant must be negative. Calculating the discriminant yields v02−4(−4.9)(−10)<0v_0^2 - 4(-4.9)(-10) < 0, which simplifies to v02−196<0v_0^2 - 196 < 0. Solving for a positive velocity gives 0<v0<140 < v_0 < 14.

Step-by-Step Solution

1
Set up the equation for the rocket reaching a height of 12 meters.
−4.9t2+v0t+2=12-4.9t^2 + v_0 t + 2 = 12
To find when the rocket reaches 12 meters, we set the height equation equal to 12.
2
Rearrange the equation into standard quadratic form.
−4.9t2+v0t−10=0-4.9t^2 + v_0 t - 10 = 0
Subtracting 12 from both sides of the equation sets it to 0, which is the standard form at2+bt+c=0at^2 + bt + c = 0.
3
Apply the condition for no real solutions using the discriminant.
Δ=v02−4(−4.9)(−10)<0\Delta = v_0^2 - 4(-4.9)(-10) < 0
For the rocket to never reach 12 meters, the quadratic equation must have no real solutions. This requires the discriminant Δ=b2−4ac\Delta = b^2 - 4ac to be strictly negative.
4
Solve the inequality for the positive velocity v0v_0.
v02−196<0  ⟹  v02<196  ⟹  v0<14v_0^2 - 196 < 0 \implies v_0^2 < 196 \implies v_0 < 14
Multiplying the constants yields 4×4.9×10=1964 \times 4.9 \times 10 = 196. Taking the square root gives v0<14v_0 < 14. Since initial upward velocity must be positive, 0<v0<140 < v_0 < 14.

Key Concept

Using the discriminant of a quadratic equation to determine the number of real solutions in a physical context.
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Topics

Quadratic Equations and the Quadratic Formula

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