document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1137680>Question 1137680</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Help with function homework.<BR>\n" );
document.write( "Gloria is skeet shooting. The height of the skeet is modelled by the function h(t)=-4.9t^2+32t+2 where h(t) is the height in metres after t seconds. The path of Gloria’s pellet is modelled by the function g(t)=28.5t+1 , with the same units. <BR>\n" );
document.write( "a) How high off the ground will the skeet be when it is hit?<BR>\n" );
document.write( "b) After how many seconds will the skeet be hit? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #755550 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=755550\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the height of the skeet is modeled by h(t) = -4.9t^2 + 32t + 2.\r<BR>\n" );
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document.write( "the path of gloria's pellet is modeled by g(t) = 28.5t + 1.\r<BR>\n" );
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document.write( "the skeet will be hit when h(t) = g(t).\r<BR>\n" );
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document.write( "it's easy to find using the desmos.com calculator.\r<BR>\n" );
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document.write( "here's what that calculator shows me.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/032911.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "the calculator says that the bullet and the skeet will meet in .933 seconds after the gun has been fired and the skeet has been launched.\r<BR>\n" );
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document.write( "based on the formulas, the gun was fired and the skeet was launched at the same time.\r<BR>\n" );
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document.write( "also based on the formulas, the skeet was launched at a height of 2 meters and the gun was fired at a height of 1 meter.\r<BR>\n" );
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document.write( "those are the y-intercepts of the grass:  the value of y when the value of x is 0.\r<BR>\n" );
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document.write( "for graphing, i replaced h(t) and g(t) with y and i replaced t with x\r<BR>\n" );
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document.write( "y still represents the height of the skeet and the bullet and x still represents the seconds from when the skeet was launched and the gun was fired.\r<BR>\n" );
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document.write( "to solve algebraically, you need to find when g(t) = h(t).\r<BR>\n" );
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document.write( "to do that, you set g(t) to h(t) and solve for t.\r<BR>\n" );
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document.write( "start with h(t) = g(t) to get -4.9t^2 + 32t + 2 = 28.5t + 1.\r<BR>\n" );
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document.write( "subtract the right side of the equation from both sides of the equation to get:\r<BR>\n" );
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document.write( "-4.9t^2 + 32t + 2 - 28.5t - 1 = 0\r<BR>\n" );
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document.write( "combine like terms to get -4.9t^2 + 30.5t + 1 = 0\r<BR>\n" );
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document.write( "use the quadratic formula to get x = -.219 of .933 rounded to 3 decimal places.\r<BR>\n" );
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document.write( "this confirms the graphic solution.\r<BR>\n" );
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document.write( "the negative value of x is discarded since the number of seconds can't be negative.\r<BR>\n" );
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document.write( "the bullet and the skeet meet in .933 seconds at a height of 27.59 seconds which you find by replacing t with .933 in either equation.\r<BR>\n" );
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document.write( "if you wanted the gun to fire after the skeet was launched, then the interception point would be different.\r<BR>\n" );
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document.write( "to do that, you replace t in the equation of g(t) = 28.5t + 1, with t-k to get g(t) = 28.5 * (t-k) + 1.\r<BR>\n" );
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document.write( "that will delay the firing of the gun for k seconds and also shift the graph of the trajectory of the bullet to the right by k seconds.\r<BR>\n" );
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document.write( "for example, assume you wanted to delay the firing of the gun for 3 seconds, then the equation for the bullet would be g(t) = 28.5 * (t - 3) + 1.\r<BR>\n" );
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document.write( "graphically, that would look like this.\r<BR>\n" );
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document.write( "the trajectory of the bullet is the same, only it's fired off at t = 3 and the intersection point of the bullet and the skeet has changed as shown on the graph.\r<BR>\n" );
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document.write( "as before, g(t) was changed to y and t was changed to x for graphing purposes.\r<BR>\n" );
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document.write( "the solution to your questions are:\r<BR>\n" );
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document.write( "a) How high off the ground will the skeet be when it is hit?\r<BR>\n" );
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document.write( "the skeet will be hit 27.591 meters high.\r<BR>\n" );
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document.write( "b) After how many seconds will the skeet be hit?\r<BR>\n" );
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document.write( "the skeet will be hit in .933 seconds. </SPAN>\n" );
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