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The height y (in feet) of a ball thrown by a child is y = c2 + 6r + 5 12 where € is the horizontal distance in feet from the point at which the ball is thrown...

Question

The height y (in feet) of a ball thrown by a child is y = c2 + 6r + 5 12 where € is the horizontal distance in feet from the point at which the ball is thrownHow high is the ball when it leaves the childs hand? Preview feet(b) What is the maximum height of the ball?PreviewfeetHow far from the child does the ball strike the ground? Preview feet

The height y (in feet) of a ball thrown by a child is y = c2 + 6r + 5 12 where € is the horizontal distance in feet from the point at which the ball is thrown How high is the ball when it leaves the childs hand? Preview feet (b) What is the maximum height of the ball? Preview feet How far from the child does the ball strike the ground? Preview feet



Answers

The height $y$ (in feet) of a baseball thrown by a child is $$y=-\frac{1}{10} x^{2}+3 x+6$$ where $x$ is the horizontal distance (in feet) from where the ball was thrown. Will the ball fly over the head of another child 30 feet away trying to catch the ball? (Assume that the child who is trying to catch the ball holds a baseball glove at a height of 5 feet.)

A clear a ball on the height of the booth. Feet can be approximated by why X We're doing my No. 0.11 exist. Bless 0.577 packs. Last fight where X is the magenta position of all measured in Pete from the video. Are they us? For what, well off heads when the ball reach its highest position. Time to the nearest now highest position in question is the vortex that's ordinate off. The verdicts is given by minus B, apparently in the even ocean a equal do my 0.11 be equal to 0.577 c do five. Substitute of A and A B medicine. Sheetal pointing to 577 doing to minus 2.1 wine be it. Does this explain to do seven, but we should ask toe around the nearest cold, so we will right? 26. No substitute experts for do 26 in a way. Get minus points. It'll one just sits a spying. You going to fight? Going? Do six plus five way. Have to a rounded to nearest rent off the food way. Get 2.6. So uncertain part is 26 foot to be. These 2.6

So no Indust Some we will substitute in this problem. Substitute X equals 2 30 for White. So therefore why is equals to minus 0.1 Took T Square plus three multiplied with 30 blessed six, which is equals toe wise equals 263 So no, no since the ball A six for you, above the widow at a distance off 30 feet. Then it will fly. What? But child who is only six feet?

Were given that the height of the ball thrown and then the air follows the given function where X represents the horizontal distance, the ball is thrown and hft represents the height or H of X represents the height. We need to find how the initial height of the ball. We need to find the maximum height and we need to find out how far from the throat does the ball hit the ground? So our initial height is when X zero. So we have a evaluated at zero. So that's negative 1 12 times zero square plus six times zero plus three. Do we have our initial height is when H zero is equal to three? So this is three feet is our initial height are maximum height occurs when we have our vertex. So we need to find our axis of symmetry. So that's going to give us our horizontal distance that the maximum height occurs. So we have X. Is equal to negative B over two. A. So is the number in front of X squared B. Is in the number in front of X. So we have negative B, which is 6/2 times negative 1 12. So we have negative six divided by two times negative 1 12. This becomes 36. So our road taxes at 36 we plug that 36 black back into the equation. So we have a evaluated at 36. We get negative 1 12 times 36 squared plus six times 36 plus three. So negative one negative 1 12 times 36 squared plus six times 36 plus three gives us a height of 111. So the max height, yeah, height Is 111 beat. So we need to find how far away it hits the ground. So we need to find our X intercept. So we're going to do the quadratic formula. Quadratic formula is X is equal to negative B plus or minus the square root. A. B squared minus four times a times C. All over two. A. So we get X is equal to negative parentheses plus minus the square root Parentheses, squared -4 times parentheses times parentheses all over two times parentheses. So C. Is the only one that was missing. So C. Is our constant at our end. So we plug in our values that we have six for B. We have negative 1 12 for A. And we've got three for C. So this simplifies to be negative six plus or minus the square root. Six square -4 times negative 1 12 times three becomes 37 over two times negative 1 12 becomes negative 16 Oops. So we're going to get one answer by doing negative six plus the square to 37 over negative 16 And we're going to get another answer equal to negative six minus the square root of 37 over negative 16 So we got to see which one is the valid answer. So we have -6 plus the square root of 37 Divided by -16. That gives us a negative answer. So this can't be the correct one. So we try with the negative sign. So negative six minus the square root of 37 Divided by -16, gives us 72.5 So it hits the ground 72.5 ft away from where it was thrown.

So we're given this position function. And first what we're doing with this position function is we're determining what the height is of the ball at its highest point. So in order to do this, we need to find where the velocity is equal to zero. Because velocity is equal to zero right about at this point because the derivative is zero the the slope of the tangent line zero. So this is really what we're looking after. We need to figure out at what time does this occur? And then we can put that time into the position function to determine the actual height. So, um, if we find the velocity of this function, we see that it is negative 32 t plus v. Not with that, um, the velocity. Now we just have to find where it's equal to zero. That's gonna be once again negative 32 t plus we not we saw for tea and see that t is going to be equal Teoh v not over 32. And we get that because we bring the 32 tea over here and then we divide by the 32 to get the T isolated So for Part B. Um, what we're going to see is that before we actually get to part B, what we have to dio is we have to now place this time into our original position function. So we now see if we evaluated at this point s a V not over a 32. This is going to equal negative 16 times. He not over 32 and that is squared, plus the not times e not over 32. And at this point, it's just simplifying to determine what this in fact is. And this is giving us the height at its maximum point. So we have 32 squared done there, and then we can cancel that out with the 16 to an extent. So it's gonna give us a negative the not squared over 64 and then So that way we can combine like terms. We're going to make this, um, 64 as well. So it's gonna us 64. And, Lord, to do that we have to double the numerator, so we will get to the not squared. And now that they have the same denominator, we can now say that the maximum height of this position function will be vey not squared, divided by 64. So that's part a, um now moving down to part B into it in green. So for part B were being asked more specifically to figure out the velocity with which the ball strikes the ground. So this we need to know when the ball strikes the ground the time at which the ball strikes the ground and when something strikes the ground, it means it has a height of zero. So what we need to figure out is, when is this going to equal zero? Okay, so we set zero equal to negative 16 t squared, plus veena t. What we can do here is we can factor out of tea. At this point. It just becomes pretty basic algebra. So I will give us a negative 16 t plus B not. And here we now have it equaling zero. So we see two cases we see t equals zero. Um, but with t equals zero, this doesn't really tell us anything when the time is equal to zero. This is the right before the ball is actually launched. So obviously the ball is not at any height because it has not yet been launched. So t equals zero we don't really care about because has nothing to do with the ball actually striking the ground because the ball hasn't even been launched yet. What we do care about is when we have zero equals negative 16 t plus me not because now we can solve for T. When we add 16 t to both sides and divide by 16 we see that t is equal to V not over 16. We're not quite done yet, though, because we just found the time in which it occurs. But what we really need to find is the velocity, um, of the ball when it strikes the ground. So we want to refer back to our velocity function, which is right here. Um, so we're gonna have v of the not over 16 and we're going to put this in place of tea when we're looking at our velocity function. So it was negative 32 t. So it is in fact, going to be negative 32 times being not over 16. And then that plus close Amina, right? And once again, just come just simple computing tells us that's going to be a negative to the not plus Vina, and that will simplify too negative v Not so. That is the velocity with which the ball actually strikes the ground.


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