document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210513>Question 1210513</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Scientists launched a rocket which can be modeled by the equation h = -16t2 + 160t + 1, where t is time in seconds, and h is the distance from the ground in feet.\r<BR>\n" );
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document.write( " a) Find the maximum height of this rocket.\r<BR>\n" );
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document.write( "____ feet\r<BR>\n" );
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document.write( "b) When does the maximum height occur?\r<BR>\n" );
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document.write( " _ seconds\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #853083 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2138)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=853083\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This problem can be solved by finding the vertex of the parabola described by the height function:\r<BR>\n" );
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document.write( "$$h(t) = -16t^2 + 260t + 1$$\r<BR>\n" );
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document.write( "This is a quadratic equation in the form $h(t) = at^2 + bt + c$, where $a = -16$, $b = 260$, and $c = 1$. Since $a$ is negative, the parabola opens downward, and the vertex represents the maximum height.\r<BR>\n" );
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document.write( "### a) Find the maximum height of this rocket.\r<BR>\n" );
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document.write( "The maximum height occurs at the time $t$ given by the axis of symmetry formula:<BR>\n" );
document.write( "$$t = -\frac{b}{2a}$$\r<BR>\n" );
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document.write( "The maximum height ($h_{\text{max}}$) is found by substituting this time back into the height equation $h(t)$.\r<BR>\n" );
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document.write( "#### Step 1: Find the time of maximum height ($t$)\r<BR>\n" );
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document.write( "$$t = -\frac{260}{2(-16)} = -\frac{260}{-32} = \frac{260}{32}$$<BR>\n" );
document.write( "$$t = 8.125 \text{ seconds}$$\r<BR>\n" );
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document.write( "#### Step 2: Calculate the maximum height ($h$)\r<BR>\n" );
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document.write( "Substitute $t = 8.125$ into the height equation:<BR>\n" );
document.write( "$$h_{\text{max}} = -16(8.125)^2 + 260(8.125) + 1$$\r<BR>\n" );
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document.write( "I will use the code interpreter to perform the calculation precisely.\r<BR>\n" );
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document.write( "```python?code_reference&code_event_index=2<BR>\n" );
document.write( "# Given function parameters<BR>\n" );
document.write( "a = -16<BR>\n" );
document.write( "b = 260<BR>\n" );
document.write( "c = 1\r<BR>\n" );
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document.write( "# Calculate time (t) of maximum height (t = -b / 2a)<BR>\n" );
document.write( "t_max = -b / (2 * a)\r<BR>\n" );
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document.write( "# Calculate the maximum height (h) by substituting t_max into the equation h = at^2 + bt + c<BR>\n" );
document.write( "h_max = a * (t_max**2) + b * t_max + c\r<BR>\n" );
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document.write( "print(f\"Time of maximum height (t_max): {t_max}\")<BR>\n" );
document.write( "print(f\"Maximum height (h_max): {h_max}\")\r<BR>\n" );
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document.write( "```\r<BR>\n" );
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document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
document.write( "Time of maximum height (t_max): 8.125<BR>\n" );
document.write( "Maximum height (h_max): 1057.25\r<BR>\n" );
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document.write( "```\r<BR>\n" );
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document.write( "### a) Find the maximum height of this rocket.\r<BR>\n" );
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document.write( "Substituting $t = 8.125$ into the equation:<BR>\n" );
document.write( "$$h_{\text{max}} = -16(8.125)^2 + 260(8.125) + 1$$<BR>\n" );
document.write( "$$\mathbf{h_{\text{max}} = 1057.25 \text{ feet}}$$\r<BR>\n" );
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document.write( "### b) When does the maximum height occur?\r<BR>\n" );
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document.write( "The maximum height occurs at the time calculated by the axis of symmetry formula:<BR>\n" );
document.write( "$$t = -\frac{b}{2a} = 8.125$$\r<BR>\n" );
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document.write( "$$\mathbf{t = 8.125 \text{ seconds}}$$\r<BR>\n" );
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document.write( "-----\r<BR>\n" );
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document.write( "**Final Answers:**\r<BR>\n" );
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document.write( "a) Find the maximum height of this rocket.<BR>\n" );
document.write( "**1057.25** feet\r<BR>\n" );
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document.write( "b) When does the maximum height occur?<BR>\n" );
document.write( "**8.125** seconds </SPAN>\n" );
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