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A rock is thrown down from the ledge of a mountain 200 feet above the ground with an initial velocity of 48 feet per second. If the height, $h,$ is given by the equation $h=-16 t^{2}-48 t+200,$ where $t$ is the time in seconds, how long does it take for the rock to hit the ground? Give your answer to the nearest one-hundredth of a second.

$$2.34 \mathrm{sec}$$

Algebra

Chapter 0

Reviewing the Basics

Section 3

Completing the Square

Equations and Inequalities

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Lectures

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A rock is dropped from the…

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Yes, this problem has a lot of words, but, um, the main thing to get out of it Uh huh. 40 T plus 200 is we're seeing when it's getting back to the earth. So the height is going to equal zero. Um, so if I were asked to do this and I'm guessing they want us to complete the square again is we want the leading coefficient to be one. So I mean, I divide everything by negative 16, which is perfectly fine and so on as we divide everything. So this will pencil 48. Divided by 16, gives me three t squared plus three t and 200 divided by a negative. 16 is 12.5 and zero. Divided by anything is still zero. So my next step, then, is I'm going to little space right here, and I'm gonna add the 12.5 to the right side. Uh, and I will instead of writing 12.5, I'll change that to a decimal 25 halves. Hope you agree that 25 halves is equal to 12.5. The reason why I like that is because then when I complete the square our processes to take this Number three divided by two squared um and you just square each piece. That's gonna be nine force. So I'm gonna add nine force to both sides. So if I were more intelligent, but I would have actually done is change this to be, uh, 50 sports Yes, 50 divided for social function. 50 forces 1200. Yeah. So the reason why I would go through that process is now I have a perfect square China Factors and Nine Force that had to be three R three halves and three have. So instead of writing three halves twice and write at once squared and on this side, now that the the nominees are the same and you just add 15 and nine together. So now I can square root both sides. I'm your skin square with each piece. And in this case, because it's a word problem, it only makes sense that a final answer will be positive. I'm going to go ahead and write, plus or minus Route 59 square before is to. But as I subtracts three halves over and with the denies being the same, I can just write negative three plus or minus Route 59. But if I did negative three minus Route 59 again answer. That's negative. And think about it. We're talking about when an object hits the ground or hits the water. Uh, we cannot get a negative. So that's why I'm in disregard that negative and just use my calculator and mega three plus Route 59 uh, and then divide that answer by two and I get an answer of 2.3 or one. I believe it's in seconds if the directions may also only want to decimals. So 2.34 But I'm gonna stop here, just circle that.

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