document.write( "Question 1157924: The transport department is collecting information about traffic levels on the SouthEast Freeway. Yesterday they placed a traffic data collection instrument at a particular location along the motorway. The instrument reported that the number of cars passing each minute varied linearly between 6:30am and 7:30am, and then quadratically (of the form at2 + bt) from 7:30am until 8am. The datashowedthat90carspassedthedatacollectorat7am, whileat7:30am150carspassedin1minute; at 8am, there were 360 cars passing in 1 minute. (The problem text probably means that 90 cars passed in the 1 minute between 7:00am and 7:01am) \r
\n" ); document.write( "\n" ); document.write( "(a) Assign a variable to be the number of cars passing the data collection instrument in any minute, and a variable to represent the number of minutes after 6:30am. \r
\n" ); document.write( "\n" ); document.write( "(b) Find the function that relates these two variables, and graph the function. \r
\n" ); document.write( "\n" ); document.write( "(c) How many vehicles were passing the data collection instrument at 6:30am? At 7:50am? \r
\n" ); document.write( "\n" ); document.write( "(d) For the linear part of the graph, explain the meaning of the slope in the context of how many cars were passing. \n" ); document.write( "
Algebra.Com's Answer #854497 by KMST(5396)\"\" \"About 
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EDIT #1: Quick edit of unintentionally posted unfinished, uncorrected answer.
\n" ); document.write( "EDIT #2: Completed missing parts of the answer.
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\n" ); document.write( "The data showed that 90 cars passed the data collector at 7am, while at 7:30am 150 cars passed in 1 minute.
\n" ); document.write( "At 8am, there were 360 cars passing in 1 minute.
\n" ); document.write( "I interpret that as the number if cars passing in 1 minute was considered as passing at 7:00am (or any other time listed) if they passed during a 1-minute period that had the exact time stated at the beginning, the end, or somewhere in the middle.
\n" ); document.write( "The text says that from 7:30am until 8am the number of cars passing per minute will vary as \"at%5E2%2Bb\" , using \"t\" as the variable to represent time. (The letters a, and b obviously represent constants)
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\n" ); document.write( "(a) Assign a variable to be the number of cars passing the data collection instrument in any minute, and a variable to represent the number of minutes after 6:30am.
\n" ); document.write( "\"t\"\"%22=%22\" number of minutes after 6:30am. (The student is invited to assign a variable, but part (b) suggest that the teacher would prefer to use \"t\" instead).
\n" ); document.write( "\"y\"\"%22=%22\" number of cars passing per minute
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\n" ); document.write( "(b) Find the function that relates these two variables, and graph the function.
\n" ); document.write( "It is obviously what they call a piecewise function:
\n" ); document.write( "linear between 6:30am and 7:30am, and then
\n" ); document.write( "quadratic (of the form at2 + bt) from 7:30am until 8am.
\n" ); document.write( "The information we have can be tabulated as
\n" ); document.write( "
\n" ); document.write( "We will need to find the constant \"a\" and \"b\" for the quadratic part of the function.
\n" ); document.write( "Linear functions are often written a of the for y=mx+b , but to avoid confusion, I would not use the letter \"b\" and say the linear part will follow the form \"y=mt%2Bp\" .
\n" ); document.write( "
\n" ); document.write( "For the linear part we have two data points:
\n" ); document.write( "\"y=90\" for \"t=30\" , and \"y=150\" for \"t=60\" .
\n" ); document.write( "From those two points we can calculate the slope as
\n" ); document.write( "\"m=%28150-90%29%2F%2860-30%29=60%2F30\"-->\"highlight%28m=2%29\"
\n" ); document.write( "Then, we can substitute into \"y=mt%2Bp\" the value found and the (t,y) values for one of the points used, to find the constant p.
\n" ); document.write( "Using point (30,90), with t=30, y=90, we get
\n" ); document.write( "\"90=3%2A20%2Bp\"-->\"90=60%2Bp\"-->\"90-60=p\"-->\"highlight%28p=30%29\"
\n" ); document.write( "NOTE: When you type that calculation, or when you key it into a calculator,
\n" ); document.write( "it must be written as \"(150-90)/(60-30)\" because
\n" ); document.write( "when we see a horizontal line separating \"150-90\" from \"60-30\",
\n" ); document.write( "we know we are supposed to calculate first the top and bottom parts,
\n" ); document.write( "but in math \"150-90/60-30\" means \"150-90%2F60-30=118.5\" ,even if your calculator has another symbol for \"divided by\".
\n" ); document.write( "
\n" ); document.write( "For the quadratic part we have two data points:
\n" ); document.write( "\"y=150\" for \"t=60\" , and \"y=360\" for \"t=90\" .
\n" ); document.write( "If we substitute each pair of values into the quadratic function \"y=at%5E2%2Bb\" , we can find a and b.
\n" ); document.write( "We get:
\n" ); document.write( "\"150=a%2A60%5E2%2Bb%2A60\"-->\"150=3600a%2B60b\" , which obviously simplifies to
\n" ); document.write( "\"15=360a%2B6b\" and further to \"5=120a%2B2b\" or \"highlight%28120a%2B2b=5%29\"
\n" ); document.write( "and
\n" ); document.write( "\"360=a%2A90%5E2%2Bb%2A90\"-->\"360=8100a%2B90b\"-->\"4=90a%2Bb\" or \"highlight%2890a%2Bb=4%29\"
\n" ); document.write( "\"system%2890a%2Bb=4%2C%22+%22%2C120a%2B2b=5%29\" is a system of linear equations in a and b.
\n" ); document.write( "Solving it, we find \"highlight%28system%28a=0.05%2C+b=-0.5%29%29\"\r
\n" ); document.write( "\n" ); document.write( "Putting both pieces of the function together, we have
\n" ); document.write( "y=\"system%282t%2B30%2C%22++%22%2C0.05t%5E2-0.5t%29\" .
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\n" ); document.write( "\n" ); document.write( "(c) How many vehicles were passing the data collection instrument at 6:30am? At 7:50am?
\n" ); document.write( "At 6:30am, \"t=0\" and we use \"y=2t%2B30\" to find \"y=2%2A0%2B30=30\"
\n" ); document.write( "At 7:50am, \"t=80\" and we use \"y=0.05t%5E2-0.5t\" to find
\n" ); document.write( "\"y=0.5%2A80%5E2-0.5%2A80=0.05%2A6400-40=320-40=280\"
\n" ); document.write( "
\n" ); document.write( "(d) For the linear part of the graph, explain the meaning of the slope in the context of how many cars were passing.
\n" ); document.write( "The slope of the linear part of the graph \"m=2\" means that each minute the number of cars passing per minute would increase by 2. \n" ); document.write( "
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