document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199547>Question 1199547</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A point travels as described by the following parametric equations x=10t+10cos3.14t, y=20t+10sin3.14t and z=30t, where x,y,z, are in meters, t in seconds, all angles in radians. The vector locating the body at any time is r=ix+jy+kz. Determine the magnitude of the velocity of the body in meters per second at time t = 0.75 second. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848287 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.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=848287\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To determine the magnitude of the velocity of the body at \( t = 0.75 \) seconds, we follow these steps:\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### **Step 1: Velocity vector**  <BR>\n" );
document.write( "The velocity vector \( \mathbf{v} \) is the time derivative of the position vector \( \mathbf{r} \):  <BR>\n" );
document.write( "\[<BR>\n" );
document.write( "\mathbf{r}(t) = x(t) \mathbf{i} + y(t) \mathbf{j} + z(t) \mathbf{k}<BR>\n" );
document.write( "\]<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "\mathbf{v}(t) = \frac{d\mathbf{r}(t)}{dt} = \frac{dx(t)}{dt} \mathbf{i} + \frac{dy(t)}{dt} \mathbf{j} + \frac{dz(t)}{dt} \mathbf{k}<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### **Step 2: Compute derivatives**<BR>\n" );
document.write( "The given parametric equations are:  <BR>\n" );
document.write( "\[<BR>\n" );
document.write( "x = 10t + 10\cos(\pi t), \quad y = 20t + 10\sin(\pi t), \quad z = 30t<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "1. Derivative of \( x \):<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "\frac{dx}{dt} = 10 - 10\pi \sin(\pi t)<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "2. Derivative of \( y \):<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "\frac{dy}{dt} = 20 + 10\pi \cos(\pi t)<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "3. Derivative of \( z \):<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "\frac{dz}{dt} = 30<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "Thus:<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "\mathbf{v}(t) = \left( 10 - 10\pi \sin(\pi t) \right) \mathbf{i} + \left( 20 + 10\pi \cos(\pi t) \right) \mathbf{j} + 30 \mathbf{k}<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### **Step 3: Compute magnitude of velocity**<BR>\n" );
document.write( "The magnitude of \( \mathbf{v} \) is:<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "|\mathbf{v}| = \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 + \left( \frac{dz}{dt} \right)^2}<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "At \( t = 0.75 \):<BR>\n" );
document.write( "1. Compute \( \sin(\pi t) \) and \( \cos(\pi t) \):<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "t = 0.75 \quad \Rightarrow \quad \pi t = 0.75\pi<BR>\n" );
document.write( "\]<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "\sin(0.75\pi) = \cos(0.25\pi) = \frac{\sqrt{2}}{2}, \quad \cos(0.75\pi) = -\sin(0.25\pi) = -\frac{\sqrt{2}}{2}<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "2. Substitute into derivatives:<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "\frac{dx}{dt} = 10 - 10\pi \cdot \frac{\sqrt{2}}{2} = 10 - 5\pi\sqrt{2}<BR>\n" );
document.write( "\]<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "\frac{dy}{dt} = 20 + 10\pi \cdot \left(-\frac{\sqrt{2}}{2}\right) = 20 - 5\pi\sqrt{2}<BR>\n" );
document.write( "\]<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "\frac{dz}{dt} = 30<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "3. Compute magnitude:<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "|\mathbf{v}| = \sqrt{\left( 10 - 5\pi\sqrt{2} \right)^2 + \left( 20 - 5\pi\sqrt{2} \right)^2 + 30^2}<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "Expand each term:<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "\left( 10 - 5\pi\sqrt{2} \right)^2 = 100 - 100\pi\sqrt{2} + 50\pi^2<BR>\n" );
document.write( "\]<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "\left( 20 - 5\pi\sqrt{2} \right)^2 = 400 - 200\pi\sqrt{2} + 50\pi^2<BR>\n" );
document.write( "\]<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "30^2 = 900<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "Add them together:<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "|\mathbf{v}|^2 = \left( 100 + 400 + 900 \right) - \left( 100\pi\sqrt{2} + 200\pi\sqrt{2} \right) + \left( 50\pi^2 + 50\pi^2 \right)<BR>\n" );
document.write( "\]<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "|\mathbf{v}|^2 = 1400 - 300\pi\sqrt{2} + 100\pi^2<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "Finally:<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "|\mathbf{v}| = \sqrt{1400 - 300\pi\sqrt{2} + 100\pi^2}<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### **Step 4: Numerical calculation**<BR>\n" );
document.write( "Using \( \pi \approx 3.1416 \):<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "|\mathbf{v}| \approx \sqrt{1400 - 300(3.1416)\sqrt{2} + 100(3.1416)^2}<BR>\n" );
document.write( "\]<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "|\mathbf{v}| \approx \sqrt{1400 - 1331.29 + 986.96} \approx \sqrt{1055.67} \approx 32.50 \, \text{m/s}<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### **Final Answer**:<BR>\n" );
document.write( "The magnitude of the velocity at \( t = 0.75 \) seconds is approximately:<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "\boxed{32.50 \, \text{m/s}}<BR>\n" );
document.write( "\] </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );