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Physics Suppose you throw a ball straight up from the ground with a velocity of 80 $\mathrm{ft} / \mathrm{s}$ . As the ball moves upward, gravity slows it. Eventually the ball begins to fall back to the ground. The height $h$ of the ball after $t$ seconds in the air is given by the quadratic function $h(t)=-16 t^{2}+80 t .$ a. How high does the ball go?b. For how many seconds is the ball in the air before it hits the ground?

Hence, the ball was for $[5 \mathrm{sec}]$ in the air before it hits the ground.

Algebra

Chapter 5

Quadratic Equations And Functions

Section 5

Quadratic Equations

Equations and Inequalities

Quadratic Functions

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University of Michigan - Ann Arbor

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all right for this problem. We have a ball that we're throwing straight up from the ground that ball has in that initial velocity of 80 feet per second's 80 feet per second. And then gravity is acting on the ball, and it's bringing it back to the ground at some point, right? So we're asked to questions. The first part is it says, How high does this ball ago thankfully, were given an equation that models the height off the ball, given the time after it was initially released so we can simply take this equation, graph it and then answer the questions that we need to answer. So I'm gonna go to does most real quick, and I've already put this equation. I just changed the tea variable tow X. That's my independent variable, and I changed h toe. Why? Which is my dependent variable? In this situation, I graphed y equals negative 16 X squared plus 80 x. The first question asked, How high does the ball go? Well, you can see here phonograph, our starting value at time. Zero. The ball is zero feet above the ground. Then it goes up, up, up, up, up, up up, but eventually gravity slows it down enough where it reaches its maximum height. We can simply click on this point, which is called the Vertex, and we can see that at X equals 2.5 A. Y is equal to 100 which means in this situation that at 2.5 seconds the ball is 100 feet above the ground. So our max height in this case, our answer for part a is 100 feet part. Be asked, How many seconds is the ball in the air before it hits the ground? Well, we can answer that question easily by looking at the graph a swell. It reaches its maximum height at 2.5 seconds, which we already said. Now, if we scroll down here, how long does it take for it to reach height equals zero again? Well, that's when it hits the X axis, which is actually just 2.5 times two for five, so the ball reaches the ground again after five seconds

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