document.write( "Question 756464: A person standing close to the edge on the top of a 170 foot building throws a baseball vertically upward. The quadratic function given below models the ball's height above the ground, s(t), in feet, t in seconds after it was thrown.
\n" ); document.write( "S(t)=-16t+64t+170\r
\n" ); document.write( "\n" ); document.write( "The ball reaches its maximum height of ___ feet after ____seconds. \n" ); document.write( "

Algebra.Com's Answer #460198 by KMST(5328) $\"\"$ $\"About$

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The equation must be
\n" ); document.write( " $\"S%28t%29=-16t%5E2%2B64t%2B170\"$
\n" ); document.write( "A quadratic equation of the form $\"y=ax%5E2%2Bbx%2Bc\"$ with $\"a%3C%3E0\"$ is the equation of a parabola in standard form. It has a minimum at the vertex if $\"a%3E0\"$ , and it has a maximum at the vertex if $\"a%3E0\"$ .
\n" ); document.write( "The equation can be written in vertex form as $\"y=a%28x-h%29%5E2%2Bk\"$ based on the coordinates of the vertex: $\"x=h\"$ and $\"y=k\"$
\n" ); document.write( "
\n" ); document.write( " $\"S%28t%29=-16t%5E2%2B64t%2B170\"$ is the equation of a parabola, and the function $\"S%28t%29\"$ has a maximum at the vertex of that parabola.
\n" ); document.write( "You may remember formulas to find the coordinates of that vertex.
\n" ); document.write( "Otherwise, you can always transform the equation into the vertex form.
\n" ); document.write( "
\n" ); document.write( "REMEMBERING FORMULAS:
\n" ); document.write( "The axis of symmetry of $\"y=ax%5E2%2Bbx%2Bc\"$ is the line $\"x=-b%2F2a\"$ and contains the vertex. In other words, thw x-coordinate of the vertex is $\"h=-b%2F2a\"$ .
\n" ); document.write( "The axis of symmetry of $\"S%28t%29=-16t%5E2%2B64t%2B170\"$ is the line $\"t=-64%2F%282%2A%28-16%29%29a\"$ --> $\"t=-64%2F%28-32%29\"$ --> $\"highlight%28t=2%29\"$ seconds.
\n" ); document.write( "That is the time-coordinate of the vertex of $\"S%28t%29=-16t%5E2%2B64t%2B170\"$ , the time when $\"S%28t%29\"$ is maximum.
\n" ); document.write( "At that time, the height of the baseball is maximum, and that height is
\n" ); document.write( " $\"S%282%29=-16%2A2%5E2%2B64%2A2%2B170\"$ --> $\"S%282%29=-16%2A4%2B128%2B170\"$ --> $\"S%282%29=-64%2B128%2B170\"$ --> $\"S%282%29=highlight%28234%29\"$ feet.
\n" ); document.write( "
\n" ); document.write( "TRANSFORMING THE EQUATION INTO VERTEX FORM:
\n" ); document.write( " $\"S=-16t%5E2%2B64t%2B170\"$ --> $\"S-170=-16t%5E2%2B64t\"$ --> $\"%28S-170%29%2F%28-16%29=%28-16t%5E2%2B64t%29%2F%28-16%29\"$ --> $\"%28S-170%29%2F%28-16%29=t%5E2-4t\"$
\n" ); document.write( "Now we can \"complete the square.\"
\n" ); document.write( " $\"%28S-170%29%2F%28-16%29=t%5E2-4t\"$ --> $\"%28S-170%29%2F%28-16%29%2B4=t%5E2-4t%2B4\"$ --> $\"%28S-170%29%2F%28-16%29%2B%28-64%29%2F%28-16%29=%28t-2%29%5E2\"$ --> $\"%28S-170-64%29%2F%28-16%29=%28t-2%29%5E2\"$ --> $\"%28S-234%29%2F%28-16%29=%28t-2%29%5E2\"$ --> $\"S-234=-16%28t-2%29%5E2\"$ --> $\"S=-16%28t-2%29%5E2%2B234\"$
\n" ); document.write( " $\"S=-16%28t-2%29%5E2%2B234\"$ is the vertex form of the equation
\n" ); document.write( "The equation $\"S=-16%28t-2%29%5E2%2B234\"$ tells you that the maximum $\"s\"$ happens
\n" ); document.write( "for $\"t=highlight%282%29\"$ seconds, when $\"S=highlight%28234%29%7D%7Dfeet%2C%0D%0Abevause+for+any+other+value+of+%7B%7B%7Bt\"$ , $\"S\"$ is $\"16%28t-2%29%5E2\"$ less than that. \n" ); document.write( "

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