document.write( "Question 2079: The height of a object in feet at time t in seconds is h= 72t-16t^2. When does it hit the ground and what is its maximum height? \n" ); document.write( "

Algebra.Com's Answer #824 by matthew_sessoms(39) $\"\"$ $\"About$

You can put this solution on YOUR website!
Good question.
\n" ); document.write( "I’m not sure what methods you are familiar with for quadratic equations. First, I will use the \"complete the square method\" (CTS) and then use the graphing method to prove myself.\r
\n" ); document.write( "\n" ); document.write( "Always write any equation in descending order. Like this, $\"-16t%5E2+%2B+72t+%2B+81+=+h\"$ . If you’re not familiar with the CTS method, write back and I’ll tell you. \r
\n" ); document.write( "\n" ); document.write( "The CTS says we must get the t^2 coefficient to be 1. We’ll do this by factoring out the -16. So we have, -16(t^2 – 9/2t) + 0\r
\n" ); document.write( "\n" ); document.write( "Next, we have to divide the -9/2 by 2 (which is -9/4) and then square that (which is 81/16). So far, we have $\"-16%28t%5E2+-+9%2F2t+%2B+81%2F16%29+%2B+81\"$ (I got the 81 by -16 * 81/16 and then changing the sign). \r
\n" ); document.write( "\n" ); document.write( "Now we have to factor the perfect square. We take half of -9/2, which is -9/4, and replace this with the -9/2 and 81/16, as well as reduce t^2 to t. So we have, -16(t – 9/4)^2 + 81\r
\n" ); document.write( "\n" ); document.write( "CTS states the equation y = a(x – h)^2 + k, where (h, k) is the maximum/minimum. Always take the opposite sign of h. Remember your h and the h in the CTS equation are not the same. Your h represents y and your t represents x. If a is negative (which yours is) then there is a max and vice versa)
\n" ); document.write( "So the maximum is (9/4, 81). Remember the direction of the x & y coordinate planes. You are looking for the height, so go with the y-coordinate, which is 81.\r
\n" ); document.write( "\n" ); document.write( "To find the time (t or x) that the object hits the ground, you must solve for t. In the original equation, we can factor out a t.
\n" ); document.write( "t(-16t + 72) = 0
\n" ); document.write( "We already know that t=0 (however, we obviously know that the object is impossible to touch the ground in 0 seconds—so just ignore this 0)\r
\n" ); document.write( "\n" ); document.write( "Solve for t in -16t + 72 = 0
\n" ); document.write( "t = -9/2\r
\n" ); document.write( "\n" ); document.write( "So, the ball reaches the maximum height of 81 ft. and it touches the ground in 4.5 seconds.\r
\n" ); document.write( "\n" ); document.write( "Here is the graph to prove my statements.
\n" ); document.write( " $\"graph%28+500%2C+400%2C+-5%2C+10%2C+-5%2C+90%2C+-16+x%5E2%2B72%2Ax%29\"$
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