Math help!!

Test Preparation
1

<p>Two ships sail from the same island port, one going north at 24 knots (nautical miles per hour) and the other east at 30 knots. The northbound ship departs at 9 AM and the eastbound ship leaves at 11 AM. How fast is the distance between them increasing at 2 PM?</p>

<p>Andy, who is 6 feet tall, is walking away from a street light pole 30 feet high at a rate of 2 feet per second. How fast is his shadow increasing in length when Andy is 24 feet from the pole?</p>

<p>The sides of an equilateral triangle are increasing at the rate of 27 inches per second. How fast is the triangle's area increasing when the sides of the triangle are each 18 inches long?</p>

<p>SO if YOU can answer any of these calculus I related rates problems, that will help my assignment that is due tomorrow! (yeah, this is an SAT thread, sorry! but please answer my question!)</p>

<p>THANKS for your help!</p>

2

<p>Related rates, I got you. I’ll go one at a time with explanations. </p>

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<p>Okay, so our basic formula is going to be the pythagorean theorem, A^2 + B^2 = C^2, since, once you’ve drawn this, you have a right triangle. </p>

<p>Our givens are: dA = 24 knots and dB = 30 knots. To find A and B in nautical miles, we multiply the rate by the time. 24 * (14 - 9) = 120 and 30 * (14 - 11) = 90, so the legs of our triangle are 120 nmi and 90 nmi respectively. To find the hypotenuse, we use the pythagorean theorem and solve for c (or recognize that this is merely a pythagorean triple of 3/4/5 multiplied by 30). Regardless, C = 150 nmi. </p>

<p>We now have reduced everything to a single variable and can take the derivative of the function using implicit differentiation. </p>

<p>f ’ (x) = 2AdA + 2BdB = 2CdC</p>

<p>f ’ (x) = 2(120)(24) + 2(90)(30) = 2(150)dC</p>

<p>f ’ (x) = 5760 + 5400 = 300dC</p>

<p>= 11160 = 300dC</p>

<p>dC = 37.2 knots</p>

3

<p>Skipping two for now since 3 is less time consuming:</p>

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<p>Given that S = 18 and dS = 27 and that the area of an equilateral triangle is:</p>

<p>sqrt(3)/4 * S^2, we have all the info we need and can just take the derivative:</p>

<p>sqrt(3)/4 * 2SdS </p>

<p>sqrt(3)/4 * 2(18)(27) </p>

<p>= 243 * sqrt(3) or 420.888 square inches</p>