Writing in mathematics

Question

what temperature would you predict for the opposite end of the rod after 1 minute, 5 minutes, 9 minutes, 17 minutes. Also write about how many minutes you would predict that the temperature would be at the end of the rod after 50 degrees, 100 degrees, 180 degrees. Finally developed a linear equation when given points. You can use words since as slope, y-intercept, slope intercept form, order pairs, etc.

time: 0 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 |

temp: 20 | 40 | 60 | 80 | 81 | 82 | 83 | 120 | 160 | 200 |

Answer 1

I plotted the data points given in blue. And the estimates in red:

An estimate for t = 1 minute would be halfway between 0 and 2 so (20 + 40)/2 = 60/2 = 30 degrees

An estimate for t = 5 minute would be halfway between 4 and 6 so (60 + 80)/2 = 140/2 = 70 degrees

An estimate for t = 9 minute would be halfway between 8 and 10 so (81 + 82)/2 = 163/2 = 81.5 degrees

An estimate for t = 17 minute would be halfway between 16 and 18 so (160 + 200)/2 = 360/2 = 180 degrees

Since 50 is halfway between 40 and 60, then an estimated time for a temp of 50 degrees would be halfway between 2 and 4 minutes so (2 + 4)/2 = 6/2 = 3 minutes

100 degrees is between 83 and 120 but not halfway. To simplify things, lets use the upper relationship to figure out the time at a Temp of 100. The slope is (200 - 120)/(18-14) = 20. The point-slope equation is y - 200 = 20(x - 18). So, plug in temp = 100 and solve for x (the time):

100 - 200 = 20x - 360

-100 = 20x - 360

260 = 20x

x = 13 minutes

Since 180 is halfway between 160 and 200, then an estimated time for a temp of 180 degrees would be halfway between 16 and 18 minutes so (16 + 18)/2 = 34/2 = 17 minutes

The linear equation is a little tricky because you must divide it into 3 pieces since there are 3 separate slopes.

For t = 0 to 6 the slope is (80 - 20)/(6 - 0) = 60/6 = 10 degrees/minute and since the y-intercept is 20, the equation is y = 10x + 20 (y is temp and x is time)

For t = 8 to 12, to slope is (83 - 81)/(12-8) = 2/4 = 1/2 so the equation, using point-slope, is y - 83 = 1/2(x - 12) or simplifying:

y - 83 = 1/2(x - 12)

y - 83 = 1/2x - 6

y = 1/2x +77

For t = 14 to 18, the slope is (200 - 120)/(18 - 14) = 80/4 = 20 degree/minute. An equation (using point-slope) is y - 200 = 20(x - 18). Simplifying:

y - 200 = 20x - 360

y = 20x - 160

The transitions between the three lines are a little messy. I'd summarize like this:

Letting Temp = y and time = x:

For 0 < x < 6: y = 10x + 20

For 6 < x < 12: y = 1/2x +77

For 12 < x < 18: y = 20x - 160