document.write( "Question 1181712: A rocket is being launched vertically over a point 𝐴 on the ground with
\n" ); document.write( "a velocity of 550 𝑚𝑖Τℎ𝑟. Twenty five miles away from point 𝐴 on the
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\n" ); document.write( "is the angle of elevation of the camera changing when the rocket
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Algebra.Com's Answer #851879 by ikleyn(52781)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "A rocket is being launched vertically over a point 𝐴 on the ground with
\n" ); document.write( "a velocity of 550 𝑚𝑖/ℎ𝑟. Twenty five miles away from point 𝐴 on the
\n" ); document.write( "ground, there is a photographer video-taping the launch. At what rate
\n" ); document.write( "is the angle of elevation of the camera changing when the rocket
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\n" ); document.write( "\n" ); document.write( "        The solution in the post by @CPhill is incorrect and his answer is incorrect,  too.\r
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\n" ); document.write( "\n" ); document.write( "        The error is in the last step,  where @CPhill converts the value of  11 mi/hr  to  radians/hr.\r
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\n" ); document.write( "\n" ); document.write( "        @CPhill mistakenly treats this value of  11 mi/hr as miles per hour.\r
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\n" ); document.write( "\n" ); document.write( "        Actually,  it is  11 radians per hour,  and conversion is just done on the way and  IS  NOT  NEEDED  anymore.\r
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\n" ); document.write( "\n" ); document.write( "        Below is my correct solution.\r
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document.write( "Here's how to solve this related rates problem:\r\n" );
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document.write( "**1. Diagram and Variables:**\r\n" );
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document.write( "*   Draw a right triangle.\r\n" );
document.write( "*   Point A is one vertex (where the rocket launches).\r\n" );
document.write( "*   The photographer is at another vertex, 25 miles away from A.\r\n" );
document.write( "*   The rocket's altitude is the vertical leg of the triangle (let's call it *y*).\r\n" );
document.write( "*   The distance from A to the photographer is the horizontal leg (25 miles).\r\n" );
document.write( "*   The angle of elevation from the photographer to the rocket is θ.\r\n" );
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document.write( "**2. Given Information:**\r\n" );
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document.write( "*   dy/dt = 550 mi/hr (rocket's velocity)\r\n" );
document.write( "*   We want to find dθ/dt when y = 25 miles.\r\n" );
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document.write( "**3. Relate Variables:**\r\n" );
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document.write( "We can relate θ and y using the tangent function:\r\n" );
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document.write( "tan(θ) = y / 25\r\n" );
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document.write( "**4. Implicit Differentiation:**\r\n" );
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document.write( "Differentiate both sides of the equation with respect to time (t):\r\n" );
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document.write( "sec²(θ) * (dθ/dt) = (1/25) * (dy/dt)\r\n" );
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document.write( "**5. Solve for dθ/dt:**\r\n" );
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document.write( "dθ/dt = (1/25) * (dy/dt) / sec²(θ)\r\n" );
document.write( "dθ/dt = (cos²(θ)/25) * (dy/dt)\r\n" );
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document.write( "**6. Find cos(θ) when y = 25 miles:**\r\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\n" );
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document.write( "**7. Substitute and Calculate:**\r\n" );
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document.write( "dθ/dt = ((1/√2)² / 25) * 550 mi/hr\r\n" );
document.write( "dθ/dt = (1/50) * 550 mi/hr = 11 radians/hr.\r\n" );
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document.write( "    +-----------------------------------------------------------------------------+\r\n" );
document.write( "    |    Conversion from mi/hr of the left side to radians/hr in the right side   |\r\n" );
document.write( "    |               is just made inside the previous formula.                     |\r\n" );
document.write( "    |         This conversion is already built into the coefficients.             |\r\n" );
document.write( "    +-----------------------------------------------------------------------------+\r\n" );
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document.write( "**Answer:** The angle of elevation is changing at a rate of 11 radians per hour when the rocket reaches an altitude of 25 miles.\r\n" );
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Post-solution note

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\n" ); document.write( "\n" ); document.write( "        11  radiance per hour is  \"%2811%2A57.2958%29%2F60\" = 10.50423  degrees per minute,\r
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\n" ); document.write( "\n" ); document.write( "        if you want to have the answer in units that are adequate to the problem.\r
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\n" ); document.write( "\n" ); document.write( "This problem was posted to this forum several years ago (about 5 years ago).\r
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\n" ); document.write( "\n" ); document.write( "It was solved by Edwin under this link\r
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\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/algebra/homework/quadratic/Quadratic_Equations.faq.question.1181710.html\r
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\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/algebra/homework/quadratic/Quadratic_Equations.faq.question.1181710.html\r
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\n" ); document.write( "\n" ); document.write( "Edwin' solution was conceptually correct,  but had an error in his implementation,  leading to incorrect answer.\r
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\n" ); document.write( "\n" ); document.write( "I found that solution via  Google search,  and placed there my corrected solution.
\n" ); document.write( "My solution follows to the  Edwin's idea/design,  but fixes/repairs that error.\r
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\n" ); document.write( "\n" ); document.write( "Now both my solutions to this problem produce the same answer,  bringing peace in your mind.\r
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