document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=51417>Question 51417</A>: <I><SPAN STYLE=\"word-break: break-all;\"> please help...thankyou\r<BR>\n" );
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document.write( "3)Suppose a baseball is shot up from the ground straight up with an initial velocity of 32 feet per second. A function can be created by expressing distance above the ground, s, as a function of time, t. This function is s = -16t2 + v0t + s0 <BR>\n" );
document.write( "16 represents 1/2g, the gravitational pull due to gravity (measured in feet per second2). <BR>\n" );
document.write( "v0 is the initial velocity (how hard do you throw the object, measured in feet per second).<BR>\n" );
document.write( "s0 is the initial distance above ground (in feet). If you are standing on the ground, then s0 = 0.\r<BR>\n" );
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document.write( "a) What is the function that describes this problem?<BR>\n" );
document.write( "Answer: \r<BR>\n" );
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document.write( "b)The ball will be how high above the ground after 1 second?<BR>\n" );
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document.write( "c)How long will it take to hit the ground?<BR>\n" );
document.write( "Answer:	<BR>\n" );
document.write( "Show work in this space. \r<BR>\n" );
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document.write( "d)What is the maximum height of the ball?  What time will the maximum height be attained?<BR>\n" );
document.write( "Answer:\r<BR>\n" );
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document.write( "Show work in this space.\r<BR>\n" );
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document.write( "4)John has 300 feet of lumber to frame a rectangular patio (the perimeter of a rectangle is 2 times length plus 2 times width). He wants to maximize the area of his patio (area of a rectangle is length times width). What should the dimensions of the patio be, and show how the maximum area of the patio is calculated from the algebraic equation.  \r<BR>\n" );
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document.write( "Show clearly the algebraic steps which prove your dimensions are the maximum area which can be obtained. Use the vertex form to find the maximum area.\r<BR>\n" );
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document.write( "Answer:	\r<BR>\n" );
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document.write( "Show work in this space.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #34355 by </B><A HREF=/tutors/aboutme.mpl?userid=AnlytcPhil><B>AnlytcPhil(1810)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=AnlytcPhil><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=AnlytcPhil><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=34355\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\"><font size = 4 color = \"green\"><b>please help...thankyou \r\n" );
document.write( "3)Suppose a baseball is shot up from the ground straight \r\n" );
document.write( "up with an initial velocity of 32 feet per second. A \r\n" );
document.write( "function can be created by expressing distance above the \r\n" );
document.write( "ground, s, as a function of time, t. This function is \r\n" );
document.write( "s = -16t² + v<sub>O</sub>t + s<sub>O</sub> \r\n" );
document.write( "<sub>O</sub>\r\n" );
document.write( "[16 represents 1/2g, the gravitational pull due to gravity \r\n" );
document.write( "(measured in feet per second²). v<sub>O</sub> is the initial velocity \r\n" );
document.write( "(how hard do you throw the object, measured in feet per second).\r\n" );
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document.write( "a) What is the function that describes this problem?\r\n" );
document.write( "Answer: \r\n" );
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document.write( ">>...initial velocity of 32 feet per second...<<\r\n" );
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document.write( ">>...v0 is the initial velocity (how hard do you throw the \r\n" );
document.write( "object, measured in feet per second)...<<\r\n" );
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document.write( "Translation:  v<sub>O</sub> = 32\r\n" );
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document.write( ">>...baseball is shot up from the ground...<<\r\n" );
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document.write( ">>...If you are standing on the ground, then s<sub>O</sub> = 0...<<\r\n" );
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document.write( "Translation:  s<sub>O</sub> = 0\r\n" );
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document.write( ">>...This function is s = -16t² + v<sub>O</sub>t + s<sub>O</sub>...<<\r\n" );
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document.write( "Translation: s = -16t² + 32t + 0\r\n" );
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document.write( "             s = -16t² + 32t  \r\n" );
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document.write( "Compare that to\r\n" );
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document.write( "          f(x) = ax² + bx\r\n" );
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document.write( "and we see that s = f(x), t = x, a = -16 and b = 32.\r\n" );
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document.write( "The vertex is the point which has coordinates\r\n" );
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document.write( "( -b/(2a), f(-b/(2a)) ) \r\n" );
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document.write( "( -(32)/(2·-16), f(-b/(2a)) )\r\n" );
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document.write( "( 1, f(1) )\r\n" );
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document.write( "f(1) = -16(1)² + 32(1) = -16 + 32 = 16\r\n" );
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document.write( "So the vertex is the point (1, 16)\r\n" );
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document.write( "This means that the maximum height of 16 feet is obtained when \r\n" );
document.write( "the elapsed time = 1 second. \r\n" );
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document.write( "b)The ball will be how high above the ground after 1 second?\r\n" );
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document.write( "             s = -16t² + 32t  \r\n" );
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document.write( "             s = -16(1)² + 32(1) = -16 + 32 = 16\r\n" );
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document.write( "c)How long will it take to hit the ground?\r\n" );
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document.write( "It is on the ground when s = 0\r\n" );
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document.write( "             s = -16t² + 32t\r\n" );
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document.write( "             0 = -16t² + 32t\r\n" );
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document.write( "    16t² - 32t = 0\r\n" );
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document.write( "Factor out 16t on the left:\r\n" );
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document.write( "    16t(t - 2) = 0\r\n" );
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document.write( "setting 16t = 0 gives t = 0\r\n" );
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document.write( "stting (t - 2) = 0 gives t = 2.\r\n" );
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document.write( "So the ball is on the ground at the start, when t = 0, and\r\n" );
document.write( "again after the ball goes up and falls back down to the ground\r\n" );
document.write( "2 seconds later.     \r\n" );
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document.write( "d)What is the maximum height of the ball?\r\n" );
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document.write( "             s = -16t² + 32t  \r\n" );
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document.write( "Compare that to\r\n" );
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document.write( "          f(x) = ax² + bx\r\n" );
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document.write( "and we see that s = f(x), t = x, a = -16 and b = 32.\r\n" );
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document.write( "The vertex is the point which has coordinates\r\n" );
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document.write( "( -b/(2a), f(-b/(2a)) ) \r\n" );
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document.write( "( -(32)/(2·-16), f(-b/(2a)) )\r\n" );
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document.write( "( 1, f(1) )\r\n" );
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document.write( "What time will the maximum height be attained?\r\n" );
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document.write( "f(1) = -16(1)² + 32(1) = -16 + 32 = 16\r\n" );
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document.write( "So the vertex is the point (1, 16)\r\n" );
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document.write( "This means that the maximum height of 16 feet is obtained when \r\n" );
document.write( "the elapsed time = 1 second.  Note that this was the same time\r\n" );
document.write( "as we were asked to use in part (b), and the 16 feet obtained\r\n" );
document.write( "in part (b) was the actual maximum height.\r\n" );
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document.write( "4)John has 300 feet of lumber to frame a rectangular patio\r\n" );
document.write( "(the perimeter of a rectangle is 2 times length plus 2 \r\n" );
document.write( "times width). He wants to maximize the area of his patio \r\n" );
document.write( "(area of a rectangle is length times width). What should \r\n" );
document.write( "the dimensions of the patio be, and show how the maximum \r\n" );
document.write( "area of the patio is calculated from the algebraic equation. \r\n" );
document.write( "Show clearly the algebraic steps which prove your dimensions \r\n" );
document.write( "are the maximum area which can be obtained. Use the vertex \r\n" );
document.write( "form to find the maximum area.\r\n" );
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document.write( "Answer: \r\n" );
document.write( "Show work in this space.\r\n" );
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document.write( "        _____L______\r\n" );
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document.write( "       |            |W\r\n" );
document.write( "       |____________|\r\n" );
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document.write( ">>...the perimeter of a rectangle is 2 times length \r\n" );
document.write( "plus 2 times width...<<\r\n" );
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document.write( "Translation:  P = 2L + 2W \r\n" );
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document.write( ">>...John has 300 feet of lumber...<<\r\n" );
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document.write( "Translation:  P = 300\r\n" );
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document.write( "So 2L + 2W = 300\r\n" );
document.write( "        2W = 300 - 2L\r\n" );
document.write( "         W = 150 - L\r\n" );
document.write( ">>...He wants to maximize the area of his patio (area of a \r\n" );
document.write( "rectangle is length times width)...<<\r\n" );
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document.write( "Translation: He wants to maximize A, where\r\n" );
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document.write( "        A = LW\r\n" );
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document.write( "Substitute (150 - L) for W:\r\n" );
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document.write( "        A = L(150 - L)\r\n" );
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document.write( "        A = 150L - L²\r\n" );
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document.write( "Arrange right side in descending powers of L.\r\n" );
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document.write( "        A = -L² + 150L\r\n" );
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document.write( "Compare that to\r\n" );
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document.write( "     f(x) = ax² + bx\r\n" );
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document.write( "and we see that y = f(x), L = x, a = -1 and b = 150.\r\n" );
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document.write( "The vertex is the point which has coordinates\r\n" );
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document.write( "( -b/(2a), f(-b/(2a)) ) \r\n" );
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document.write( "( -(150)/(2·-1), f(-b/(2a)) )\r\n" );
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document.write( "( 75, f(75) )\r\n" );
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document.write( "f(75) = -1(75)² + 150(75) = -1(5625) + 11250 = 5625\r\n" );
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document.write( "So the vertex is the point (75, 5625)\r\n" );
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document.write( "This means that the maximum area pf 5625 square feet is obtained when \r\n" );
document.write( "the length L = 75.  Substituting 75 for L in\r\n" );
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document.write( "         W = 150 - L  \r\n" );
document.write( "         W = 150 - 75\r\n" );
document.write( "         W = 75\r\n" );
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document.write( "This means that the maximium area of 5625 square feet is obtained when\r\n" );
document.write( "the Length and the width are both 75 feet, which means it is square.\r\n" );
document.write( "My, that's a big patio!\r\n" );
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document.write( "Edwin</PRE>       </SPAN>\n" );
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