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The height of a ball that is thrown directly upward from a point 200 feet above the ground with an initial velocity of 40 feet per second is given by $h(t)=-16 t^{2}+40 t+200,$ where $t$ is the amount of time elapsed since the ball was thrown; $t$ is in seconds and $h(t)$ is in feet. For what values of $t$ will the height of the ball be below 100 feet?

\(t\) \(\geq\) \(4\) seconds

Precalculus

Chapter 2

More About Functions and Equations

Section 7

Quadratic Inequalities

Trigonometry

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all upward and we want to know at what time is a below a certain certain. So we want to know where this ball is below 100 feet if we threw it at 200 feet. And so now we are given a a height function. But we want to know where that height is less than 100. And so if we have the quadratic or we have the function of the height, we want to know where it's less than 100. We can figure out where where we can. We can manipulate that and say OK, so we have we have negative 16 t squared negative 16. You have 16 t squared plus 40 t and then since we subtracted 100 to get it to make make the height lessons here is that we have plus ah, 100 is less than zero feet. And so now we can solve using the quadratic formula. So we have negative 40 plus or minus plus or minus the square root of B squared, which is 16. It ISS 16 100 minus for times negative. 16 4 times negative 16 times 100. So that's plus 6400 plus 60 or 100 Ah, 6400 over two times Negative 16. Negative, 30 to. And so if we were to solve for this, we would get we would get the root of the square root of 1600 plus 6400 is the same thing as saying is the same thing of single the square of 8000. So the square root of 8000 is 89.442 But if we say negative for you, plus this value. So if we were to softens, we can we can figure out the values of T. It's a let's a T is equal to well, we have negative 40 plus 89.442 which is a square of 8000 that and then provided by negative 32. That leaves us with negative 1.5 negative 1.5 and the other instance in which case we subtract negative 40 minus that value negative. 40 minus the square root of 8000 we're left with. We're left with 4.4 so four point zero for Or let's keep it at four points here or four seconds, so it's round to the nearest whole number or point. But let's round two, the newer stencil. So we have We have two instances. We have a negative time and we have a positive time. So we already know that in real world applications, negative time doesn't really we don't worry about that. So, missus is we have t is equal to four. And so if we were to draw a graph of this, we have at at time T equals zero Time T equals zero were at some value 200 and we go down. And so we want to know we want to know Where is this? Where is this? Less than some value. And so if that occurs at four seconds, T equals four seconds. We know that when t that when t is greater than for that is when our height will be less than 100 feet

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