A flare is fired from the bottom of a gorge with an initial velocity of 144 feet per second, and it is visible only when the flare is above the rim. The height of the flare, in feet, after $t$ seconds is given by the function $h(t)=-16 t^{2}+144 t$. If the gorge is 288 feet deep, for what length of time can the flare be seen?
A) $6 \mathrm{sec}$ to $9 \mathrm{sec}$
B) $3 \mathrm{sec}$ to $6 \mathrm{sec}$
C) $0 \mathrm{sec}$ to $3 \mathrm{sec}$
D) $9 \mathrm{sec}$ to $12 \mathrm{sec}$

Question

Quizplus · Accepted Answer

To find the time when the flare can be seen, we need to determine when its height is above 0 feet (since the gorge is 288 feet deep, but the question effectively sets the rim as the height reference point). Setting the height function equal to 0: $-16t^2 + 144t = 0$. Factoring out $t$, we get $t(-16t + 144) = 0$, which gives $t = 0$ and $t = 9$. This means the flare is at ground level at $t = 0$ and $t = 9$ seconds. Therefore, it is visible from $0$ seconds to $9$ seconds. However, since it starts at the bottom of the gorge, it actually becomes visible once it rises above the rim. Assuming the question implies visibility starts once it's launched, it's visible from the start. The correct interpretation, given the depth of the gorge isn't directly factored into the visibility calculation, is that it's visible from launch until it lands, hence from $0$ to $9$ seconds. However, the correct choice should reflect the time it's above the rim, which is immediately upon launch until it peaks and falls back down, making the correct answer from $0$ seconds to $9$ seconds, but since that option isn't provided directly and considering the depth of the gorge might be a misinterpretation in the explanation, the focus should be on the time it's in the air based on the quadratic equation's solution. Given the misunderstanding in the initial explanation, the correct interpretation based on the provided options and the quadratic equation's solution would be the interval it's above the ground level, which is from $0$ to $9$ seconds, but the visibility in terms of being above the rim isn't directly questioned; it's about the time it's in flight and visible. The options might have been misinterpreted in the explanation provided, leading to confusion. The correct answer, based on the visibility from when it's launched until it returns to the rim level, aligns with the time it's in flight, which is from $0$ to $9$ seconds, but given the options and re-evaluating the explanation, the focus should be on the visibility in terms of the flight time, which is from launch until it lands, hence the entire duration it's in the air.

A Flare Is Fired from the Bottom of a Gorge ttt

A Flare Is Fired from the Bottom of a Gorge $t$