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Problem 41 Hard Difficulty

A hawk flying at 15 m/s at an altitude of 180 m accidentally drops its prey. The parabolic trajectory of the falling prey is described by the equation
$$ y = 180 - \frac{x^2}{45} $$
until it hits the ground, where $ y $ is its height above the ground and $ x $ is the horizontal distance traveled in meters. Calculate the distance traveled by the prey from the time it is dropped until the time it hits the ground. Express your answer correct to the nearest tenth of a meter.

Answer

$45 \sqrt{17}+\frac{45}{4} \ln (4+\sqrt{17}) \approx 209.1 \mathrm{m}$

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Video Transcript

he had square sewing you marine here. So it's given that the praying is dropped from 180 meters, which is excess equal to zero. So we have to find when y is equal to zero. So we have zero is equal to 1 80 minus x squared over 45 and this gives us an X value of 90. So the distance traveled was darkling from when x is equal to zero until X is equal to 90. So next we're gonna find the derivative and we have our equation. Why is equal to 1 80 minus x square over 45? So we got negative to over 45 bucks and we just have to square this to get, and then we put this in to our or click formula. So this becomes too square over 45 square times x square, 0 to 80 square root of one plus two square over 45 square. Next square, the ex. We're gonna make me x equal to 45 over too tangent. And that means that D X is equal to 45 over to seek in square DT. So l is equal to from A to B square root one plus two square over 45 square times 45 over too Tangent square 45 over too. Sequence square. DT. This becomes equal to 45 over too from a to B square root of one plus 10 gin square. So seek in square deep team which is equal to 45 over too into girl from a to B square root of sequins square seeking square P. D. T. Is equal to 45 over too integral from a toothy seeking cube. 18. We're gonna do integration by parts. So we have you. He's second, which makes do you equal to seek it. Times tangent d team. We're gonna make Devi be equal to seek in square DT, which makes me be called to tangent. So when we do integration my parts we have from me to be seeking cubed P. D t. It was equal to U V minus integration of me. Do you? This is becomes equal to seek int times tangent minus the integral of seeking cubed 18 plus the integral of second d T. And we have to add seeking cube to both sides and we divide by two so that gives us seeking Q d. T is equal to 1/2 seeking times tangent plus 1/2 Ellen. Well, second plus tangent. And then now we're gonna solve for l. So we substituted X as equal to 45 over to 10. So that means that T is equal to worked 10 of to over 45 x. It's only fined for Integra. Lt's A is going to be equal to the ark. 10 of two over 45 times zero. So that just gives us zero. And for B, it's gonna be to over 45 times 90. And we're finding the ark 10 of that. So that's our 10 of four. So and then we just plug this in. When we find for that we get around 209 0.1 meters.