Difference between revisions of "Heating of a liquid"

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A liquid with dissolved solids was placed on a stove at a certain setting and the temperature of the liquid was measured over time. The data was graphed (temperature vs time) which gave a curved line that maxed out. A best fit of the data yielded an equation for the line, and the first derivative of that equation gave a rate of change of the temperature per unit time:
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A liquid with dissolved solids was placed on a stove at a certain setting and the temperature of the liquid was measured over time. The data was graphed (temperature vs time) which gave a curved line that maxed out. A best fit of the data yielded an equation for the line, and the first derivative of that equation gave a rate of change of the temperature per unit time at each time point:
  
f(t)=30e<sup>−0.3t</sup>
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f '(t)=30e<sup>−0.3t</sup>
  
This equation describes how the liquid responds to the heat setting of the stove. In this case, f(t) is the temperature change per minute and t is the time in minutes.
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This equation describes how the liquid responds to the heat setting of the stove.
  
For this experiment, the amount that the temperature increased from 0 to 5 minutes can be calculated using integrals, which can give total amounts.
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For this experiment, the amount that the temperature increased from 0 to 5 minutes can be calculated using integrals, which can give total amounts:
  
<font size = "+2"><span>&#8747;</span></font>f(t)dt = <font size = "+2"><span>&#8747;</span></font>30e<sup>−0.3t</sup>dt
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<font size = "+2"><span>&#8747;</span></font>f '(t)dt = <font size = "+2"><span>&#8747;</span></font>30e<sup>−0.3t</sup>dt
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= (-30/0.3)(e<sup>−0.3*5</sup> - e<sup>0.3*0</sup>) this result is found by looking up the integral <font size = "+2"><span>&#8747;</span></font>ae<sup>−bt</sup>dt in a table of integrals (or [https://www.integral-calculator.com/ here]) and using the boundary conditions 0 and 5.
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= 77.7 degrees
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in conclusion, integrating a rate of change equation will give the total change!

Latest revision as of 17:14, 1 April 2021

A liquid with dissolved solids was placed on a stove at a certain setting and the temperature of the liquid was measured over time. The data was graphed (temperature vs time) which gave a curved line that maxed out. A best fit of the data yielded an equation for the line, and the first derivative of that equation gave a rate of change of the temperature per unit time at each time point:

f '(t)=30e−0.3t

This equation describes how the liquid responds to the heat setting of the stove.

For this experiment, the amount that the temperature increased from 0 to 5 minutes can be calculated using integrals, which can give total amounts:

f '(t)dt = 30e−0.3tdt

= (-30/0.3)(e−0.3*5 - e0.3*0) this result is found by looking up the integral ae−btdt in a table of integrals (or here) and using the boundary conditions 0 and 5.

= 77.7 degrees

in conclusion, integrating a rate of change equation will give the total change!