document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1181712>Question 1181712</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A rocket is being launched vertically over a point 𝐴 on the ground with<BR>\n" );
document.write( "a velocity of 550 𝑚𝑖Τℎ𝑟. Twenty five miles away from point 𝐴 on the<BR>\n" );
document.write( "ground, there is a photographer video-taping the launch. At what rate<BR>\n" );
document.write( "is the angle of elevation of the camera changing when the rocket<BR>\n" );
document.write( "achieves an altitude of 25 miles? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849977 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849977\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve this related rates problem:\r<BR>\n" );
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document.write( "**1. Diagram and Variables:**\r<BR>\n" );
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document.write( "*   Draw a right triangle.<BR>\n" );
document.write( "*   Point A is one vertex (where the rocket launches).<BR>\n" );
document.write( "*   The photographer is at another vertex, 25 miles away from A.<BR>\n" );
document.write( "*   The rocket's altitude is the vertical leg of the triangle (let's call it *y*).<BR>\n" );
document.write( "*   The distance from A to the photographer is the horizontal leg (25 miles).<BR>\n" );
document.write( "*   The angle of elevation from the photographer to the rocket is θ.\r<BR>\n" );
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document.write( "**2. Given Information:**\r<BR>\n" );
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document.write( "*   dy/dt = 550 mi/hr (rocket's velocity)<BR>\n" );
document.write( "*   We want to find dθ/dt when y = 25 miles.\r<BR>\n" );
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document.write( "**3. Relate Variables:**\r<BR>\n" );
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document.write( "We can relate θ and y using the tangent function:\r<BR>\n" );
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document.write( "tan(θ) = y / 25\r<BR>\n" );
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document.write( "**4. Implicit Differentiation:**\r<BR>\n" );
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document.write( "Differentiate both sides of the equation with respect to time (t):\r<BR>\n" );
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document.write( "sec²(θ) * (dθ/dt) = (1/25) * (dy/dt)\r<BR>\n" );
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document.write( "**5. Solve for dθ/dt:**\r<BR>\n" );
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document.write( "dθ/dt = (1/25) * (dy/dt) / sec²(θ)<BR>\n" );
document.write( "dθ/dt = (cos²(θ)/25) * (dy/dt)\r<BR>\n" );
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document.write( "**6. Find cos(θ) when y = 25 miles:**\r<BR>\n" );
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document.write( "When y = 25 miles, the triangle is a right isosceles triangle, so θ = 45 degrees or π/4 radians.  Therefore, cos(θ) = cos(45°) = 1/√2.\r<BR>\n" );
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document.write( "**7. Substitute and Calculate:**\r<BR>\n" );
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document.write( "dθ/dt = ((1/√2)² / 25) * 550 mi/hr<BR>\n" );
document.write( "dθ/dt = (1/50) * 550 mi/hr = 11 mi/hr\r<BR>\n" );
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document.write( "**8. Convert to radians per hour:**\r<BR>\n" );
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document.write( "Since angular velocity is typically expressed in radians per unit time, we need to convert miles/hour to radians/hour. The relationship between arc length (s), radius (r), and angle (θ in radians) is s = rθ. If we consider the distance from the observer to the launch point as the radius (25 miles), and the altitude of the rocket as the arc length (25 miles), then when the altitude is 25 miles, the angle of elevation is π/4 radians. The rate of change of the rocket's altitude is given in miles per hour. The rate of change of the angle of elevation will be in radians per hour.\r<BR>\n" );
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document.write( "When y = 25 miles, the angle of elevation is 45 degrees or π/4 radians.\r<BR>\n" );
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document.write( "dθ/dt = 11 miles/hour * (1 radian / 25 miles) = 0.44 radians/hour.\r<BR>\n" );
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document.write( "**Answer:** The angle of elevation is changing at a rate of 0.44 radians per hour when the rocket reaches an altitude of 25 miles.<BR>\n" );
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