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On a reality TV show Last Man Standing, a package is thrown from a plane to the ocean below. The contestants must swim to the package to receive the "free pass" located inside the package. The path that the package follows can be modelled by the quadratic functino d(t)= -4.9t2+10t+1200, where d represents the distance, in metres, that the package travels and t is the time in seconds. How long will it take for the package to reach the water?

5 years ago

Answered By Alex M

Before we actually start doing any math, it's worth trying to think about what the question is actually describing. We have an equation for distance in terms of time, meaning that for any given time, we can find a distance. Or, to put it differently, time will be our X axis and distance will be our Y axis. Because we're given a quadratic equation, it might be tempting to quickly punch it into our graphing calculator to get an idea of what is going on, and I've included a graph below. Even without actually putting anything into our graphing calculator though, we can figure a few things out. Firstly, we're dealing with a parabola (because we have an x2 term), and it opens downwards (because the x2 term is negative).Secondly, that parabola is going to starting REALLY high up, since we're adding 1,200 to it. Finally, we have to think about what we actually are trying to figure out. We want the time when the package hits the water. The question isn't exactly clear about this, but it's perfectly reasonable to assume in questions dealing with "the ground" or "the bottom" or something like that that "the ground" is defined as the place where the distance is equal to zero. Thinking about what our parabola actually looks like, that makes sense. We have a parabola with an origin REALLY high up that opens downwards (think something like an upside-down U) and we need to find where it intersects the x axis – in other words where y is zero.That's great news, as we can set our equation as equal to zero and solve! Unfortunately, we don't have easy numbers here, so we just have to do calculator work. Our first step should look like this: $\frac{-10\pm\sqrt{10^2-4\left(-4.9\right)\left(1200\right)}}{2\left(-4.9\right)}$10±1024(4.9)(1200)2(4.9)  

Simplifying this, we get  $\frac{-10\pm153.69}{-9.8}$10±153.699.8  

Here, we get two possible answers, one that is roughly equal to 16.70 and another equal to roughly -14.66. One last step! We have to figure out which of these answers makes sense (or if both do). Since this is an equation that expresses distance in terms of time, it probably doesn't make sense to think about time in terms of negatives. The question is asking us about something that is going to happen in the future, so we expect the correct answer to have a positive time rather than a negative one. So our answer is 16.7 seconds

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