document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1194267>Question 1194267</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a laboratory experiment, it is observed that when the temperature is reduced 𝑇 (degrees Celsius) of a rabbit, its heart rate (beats per minute) decreases. Under laboratory conditions, a rabbit at a temperature of 37°𝐶 has a heart rate of 220 and at a temperature of 32°𝐶 its heart rate heart rate decreases to 150. The function 𝑅 is heart rate and is linearly related to the temperature 𝑇. a) Find the functional relationship between 𝑅 and 𝑇 and its respective graph. b) Find the variation of 𝑅 if 𝑇 = [26, 38]. </SPAN>\n" );
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document.write( "Part (a)\r<BR>\n" );
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document.write( "Given info:<BR>\n" );
document.write( "Temperature = 37 degrees C ---> heartrate = 220 bpm (beats per minute)<BR>\n" );
document.write( "Temperature = 32 degrees C ---> heartrate = 150 bpm \r<BR>\n" );
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document.write( "R = heart rate in bpm<BR>\n" );
document.write( "T = temperature in degrees C\r<BR>\n" );
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document.write( "Since R is linearly related to T, this means a straight line goes through the two points (37, 220) and (32,150)<BR>\n" );
document.write( "Each point is of the form (T, R) with T being the input independent variable and R is the output dependent variable.\r<BR>\n" );
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document.write( "Let <BR>\n" );
document.write( "x = T<BR>\n" );
document.write( "y = R<BR>\n" );
document.write( "So each point is now of the form (x,y)\r<BR>\n" );
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document.write( "Let's find the equation y = mx+b<BR>\n" );
document.write( "We'll first need the slope m<BR>\n" );
document.write( "m = slope<BR>\n" );
document.write( "m = rise/run<BR>\n" );
document.write( "m = (change in y)/(change in x)<BR>\n" );
document.write( "m = (y2-y1)/(x2-x1)<BR>\n" );
document.write( "m = (150-220)/(32-37)<BR>\n" );
document.write( "m = -70/(-5)<BR>\n" );
document.write( "m = 14<BR>\n" );
document.write( "The slope is 14 to indicate each time the temperature increases by 1 degree, the heart rate goes up by 14 bpm.<BR>\n" );
document.write( "In other words, it increases by 14 bpm per degree.\r<BR>\n" );
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document.write( "Now find the y intercept<BR>\n" );
document.write( "We'll use the slope we just found and let's say the point (x,y) = (37,220)<BR>\n" );
document.write( "y = mx+b<BR>\n" );
document.write( "y = 14x+b<BR>\n" );
document.write( "220 = 14*37+b<BR>\n" );
document.write( "220 = 518+b<BR>\n" );
document.write( "220-518 = b<BR>\n" );
document.write( "-298 = b<BR>\n" );
document.write( "b = -298<BR>\n" );
document.write( "This is the bpm when the temperature is 0 degrees C. <BR>\n" );
document.write( "Of course it's not realistic because we can't have a negative bpm. However, it does help set up the line. We'll just need to be careful on how to set up the domain.\r<BR>\n" );
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document.write( "Since m = 14 is the slope and b = -298 is the y intercept, we go from y = mx+b to y = 14x-298<BR>\n" );
document.write( "Then that leads to R = 14T - 298\r<BR>\n" );
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document.write( "Let's see what happens when we plug in T = 37<BR>\n" );
document.write( "R = 14T - 298<BR>\n" );
document.write( "R = 14(37) - 298<BR>\n" );
document.write( "R = 518 - 298<BR>\n" );
document.write( "R = 220<BR>\n" );
document.write( "As expected, a temperature of 37 degrees C leads to a heart rate of 220 bpm.\r<BR>\n" );
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document.write( "Now try T = 32<BR>\n" );
document.write( "R = 14T - 298<BR>\n" );
document.write( "R = 14(32) - 298<BR>\n" );
document.write( "R = 448 - 298<BR>\n" );
document.write( "R = 150<BR>\n" );
document.write( "This works out as well. A temperature of 32 degrees C leads to a heart rate of 150 bpm.<BR>\n" );
document.write( "This paragraph and the previous one helps fully confirm we have the correct linear function to connect R and T together.\r<BR>\n" );
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document.write( "Answer: <font color=red>R = 14T - 298</font>\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "I'm not sure what you mean here. I think something got lost in translation. \r<BR>\n" );
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document.write( "Though if you wanted to know the range of bpm values for the given temperatures, then,\r<BR>\n" );
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document.write( "Plug in T = 26<BR>\n" );
document.write( "R(T) = 14T - 298<BR>\n" );
document.write( "R(26) = 14(26) - 298<BR>\n" );
document.write( "R(26) = 66\r<BR>\n" );
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document.write( "Do the same for T = 38<BR>\n" );
document.write( "R(T) = 14T - 298<BR>\n" );
document.write( "R(38) = 14(38) - 298<BR>\n" );
document.write( "R(38) = 234\r<BR>\n" );
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document.write( "Find the difference:<BR>\n" );
document.write( "R(38) - R(26) = 234 - 66 = 168<BR>\n" );
document.write( "The range of bpm values is 168 when considering the temperature interval [26, 38] aka <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=26+%3C=+T+%3C=+38 ALIGN=MIDDLE  ALT=\"26+%3C=+T+%3C=+38\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Once again, I'm not sure what you're asking for this part (b). <BR>\n" );
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