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Difficulty: Very hardQuadratic Equations and the Quadratic Formula

A projectile is launched from ground level with an initial upward velocity of vv meters per second. The height of the projectile in meters after tt seconds is modeled by the equation h(t)=−92t2+vth(t) = -\frac{9}{2}t^2 + vt. A sensor is placed at a height of 3232 meters. What is the minimum integer value of vv for which the projectile is at or above the height of the sensor for a duration of at least 44 seconds?

Answer: 30 meters per second

Answer

The minimum integer value of the initial upward velocity vv is 30.
The height condition h(t)≥32h(t) \ge 32 leads to the quadratic inequality 9t2−2vt+64≤09t^2 - 2vt + 64 \le 0. The roots t1t_1 and t2t_2 of 9t2−2vt+64=09t^2 - 2vt + 64 = 0 represent the boundary times. The duration above 3232 meters is the difference of the roots: t2−t1=b2−4ac∣a∣=4v2−23049t_2 - t_1 = \frac{\sqrt{b^2 - 4ac}}{|a|} = \frac{\sqrt{4v^2 - 2304}}{9}. Setting this to be at least 44 seconds gives 4v2−23049≥4\frac{\sqrt{4v^2 - 2304}}{9} \ge 4. Squaring both sides after multiplying by 99 yields 4v2−2304≥12964v^2 - 2304 \ge 1296, which simplifies to 4v2≥36004v^2 \ge 3600, or v2≥900v^2 \ge 900. Taking the positive square root gives v≥30v \ge 30, meaning the minimum integer value is 3030.

Step-by-Step Solution

1
Write the inequality representing the condition that the projectile is at or above the sensor height.
9t2−2vt+64≤09t^2 - 2vt + 64 \le 0
The height function h(t)=−92t2+vth(t) = -\frac{9}{2}t^2 + vt must be greater than or equal to 3232 meters.
2
Use the formula for the difference of roots to represent the time duration.
t2−t1=4v2−23049t_2 - t_1 = \frac{\sqrt{4v^2 - 2304}}{9}
The duration of time the projectile is above the sensor is the difference between the two times at which the projectile crosses the sensor height.
3
Set the duration to be at least 44 seconds and solve for vv.
v≥30v \ge 30
Solving the inequality 4v2−23049≥4\frac{\sqrt{4v^2 - 2304}}{9} \ge 4 yields the range of initial velocities.
4
Find the smallest integer in the solution set.
3030
Since the velocity must be an integer and at least 3030, the minimum possible integer value is 3030.

Key Concept

Solving quadratic inequalities and finding the difference of roots using the discriminant.
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Topics

Quadratic Equations and the Quadratic Formula

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