document.write( "Question 115783This question is from textbook Algebra For college students
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\n" ); document.write( "\n" ); document.write( "These two problems are really hard can anyone help. \n" ); document.write( "
Algebra.Com's Answer #716009 by ikleyn(52781)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "1.  You may often meet these problems on a projectile thrown vertically upward.\r\n" );
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document.write( "    The equation for the height over the ground usually has ONE OF TWO POSSIBLE forms:\r\n" );
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document.write( "        a)  h(t) = -16*t^2 + v*t + c\r\n" );
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document.write( "                 In this form, the equation is written for the height h(t) over the ground measured in feet.\r\n" );
document.write( "                 The value of \"16\" is the half of the value of the gravity acceleration g = 32 ft/s^2.\r\n" );
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document.write( "                 The sign \"-\" at the first term means that the gravity acceleration is directed down, \r\n" );
document.write( "                 while the \"y\"-axis of the coordinate system is directed vertically up, in the opposite direction.\r\n" );
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document.write( "                 The value of \"v\" in this equation is the value of the initial vertical velocity. \r\n" );
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document.write( "                 The value of \"c\" is the initial height over the ground.\r\n" );
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document.write( "                 The ground level is assumed to be 0 (zero, ZERO). In other words, the origin of the coordinate system is at the ground.\r\n" );
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document.write( "        b)  h(t) = -5*t^2 + v*t + c\r\n" );
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document.write( "                 It is another form of the \"height\" equation for the same process.\r\n" );
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document.write( "                 In this form, the height h(t) is measured in meters (instead of feet).\r\n" );
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document.write( "                 The value of \"5\" at the first term is the same gravity acceleration, but this time expressed in \"m/s^2\" units\" g = 10 m/s^2.\r\n" );
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document.write( "                       Actually, more precise value is g = 9.8 m/s^2, therefore, sometimes, this equation goes with the first term -4.9.\r\n" );
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document.write( "                 The value of \"v\" is the vertical velocity, expressed in m/s.\r\n" );
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document.write( "                 The value of \"c\" is the initial height over the ground in meters.\r\n" );
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document.write( "2.  In any case, when such problems come from Algebra (as Algebra problems), they are treated in THIS WAY:\r\n" );
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document.write( "    The question \"find the maximal height\" is the same as \"find the maximum of the quadratic form h(t) = -16t^2 + vt + c.\r\n" );
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document.write( "    It doesn't matter that the quadratic function presented as the function of \"t\" instead of more usual \"x\" variable.\r\n" );
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document.write( "    Next, when the question is about the maximum/minimum of a quadratic form \r\n" );
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document.write( "    q(x) = ax^2 + bx + c,\r\n" );
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document.write( "    the Algebra teaches us that the maximum is achieved at   x = \"-b%2F%282a%29\".\r\n" );
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document.write( "    \r\n" );
document.write( "    In your case this value of \"t\", which provides the maximum height, is  t = \"-308%2F%282%2A%28-16%29%29\" = 9.625 seconds.     \r\n" );
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document.write( "3.  Same problems may come from CALCULUS.  In Calculus, they are treated in this way:\r\n" );
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document.write( "    to find the maximum (minimum), take the derivative and equate it to zero. \r\n" );
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document.write( "    It will give the equation to find \"t\".\r\n" );
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document.write( "4.  Same problems may come from PHYSICS.  In Physics, they are treated in this way:\r\n" );
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document.write( "    the maximum height is achieved when the verical velocity becomes equal to zero. \r\n" );
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document.write( "    It will give the equation to find \"t\":   t = \"v%2Fg\".\r\n" );
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document.write( "5.  The amazing fact is that different approaches from different branches of Math and Science give the same answer.\r\n" );
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document.write( "    AMAZING ? - Yes, of cource, without doubts for young students.\r\n" );
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document.write( "    AMAZING ? - Yes, but not so much for more mature students, who understand that all these branches of knowledge \r\n" );
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document.write( "                study the same Nature's phenomenos.  So, the results should (and must) be identical.\r\n" );
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\n" ); document.write( "On finding maximum/minimum of a quadratic function see my lessons \r
\n" ); document.write( "\n" ); document.write( "    - HOW TO complete the square to find the minimum/maximum of a quadratic function\r
\n" ); document.write( "\n" ); document.write( "    - Briefly on finding the minimum/maximum of a quadratic function\r
\n" ); document.write( "\n" ); document.write( "    - HOW TO complete the square to find the vertex of a parabola\r
\n" ); document.write( "\n" ); document.write( "    - Briefly on finding the vertex of a parabola\r
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\n" ); document.write( "\n" ); document.write( "My lessons on a projectile thrown/shot/launched vertically up are\r
\n" ); document.write( "\n" ); document.write( "    - Problem on a projectile moving vertically up and down\r
\n" ); document.write( "\n" ); document.write( "    - Problem on an arrow shot vertically upward\r
\n" ); document.write( "\n" ); document.write( "    - Problem on a ball thrown vertically up from the top of a tower \r
\n" ); document.write( "\n" ); document.write( "    - Problem on a toy rocket launched vertically up from a tall platform\r
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\n" ); document.write( "\n" ); document.write( "Also,  you have this free of charge online textbook in ALGEBRA-I in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this textbook under the topic  \"Finding minimum/maximum of quadratic functions\"\r
\n" ); document.write( "\n" ); document.write( "and under the topic  \"Projectiles launched/thrown and moving vertically up and dawn\".\r
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\n" ); document.write( "\n" ); document.write( "Save the link to this online textbook together with its description\r
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\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-I
\n" ); document.write( "https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson\r
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