function equation

Question

At the Freeze-O-Matic test facility, there is a machine that can quickly adjust temperatures in a test room. The room starts at zero degrees. The temperature drops at a rate of 10 degrees per minute for the first 8 minutes. Then, the lab techs lower the cooling rate to 10 degrees every 4 minutes for the next 10 minutes. Finally the lab techs lower the cooling rate a final time to 10 degrees for every 8 minutes for the remainder of the test time. Write a function you can use to calculate the temperatures and the fill in the table below. Round to the nearest 10th.

Answer 1

So it looks like there are three scenarios, so we will have a piecewise function.

#1: Temperature drop 10° in 1 minute (slope is -10) from 0 minutes to 8 minutes (0 < x ≤ 8)

#2: Temperature drop 10° in 4 minutes (slope is -2.5) from 8 minutes to 18 minutes (8 < x ≤ 18)

#3: Temperature drop 10° in 8 minutes (slope is -1.25) anytime after 18 minutes (x > 18)

We have to find the equations for each scenario

#1: Beginning at 0°, our slope is -10 so the equation is y = -10x ending when x = 8 at point (8,-80)

#2: Beginning at (8,-80), our slope is -2.5 so we have to use the point slope form of a line to find the equation:

y+80 = -2.5(x-8)

y = -2.5x+20-80 or y = -2.5x - 60 This scenario ends when x = 18 at point (18,-105)

#3: Beginning at (18, -105) our slope is -1.25 and we have to use the point slope form of a line to find the equation:

y+105 = -1.25(x-18)

y = -1.25x+22.5-105 or y = -1.25x - 82.5 This scenario continues after 18 minutes.

So our final piecewise function will be

-10x if 0 < x ≤ 8

f(x) = -2.5x - 60 if 8 < x ≤ 18

-1.25x - 82.5 if x > 18

To find the temperature, choose one of these three functions based on the x value (minutes) you are plugging in.

If the minutes are from 0 through 8 use the first equation

If the minutes are from 8 through 18 use the second equation

If the minutes are greater than 18, use the third equation

Hope this helps.