April 11th, 2017, 07:11 PM  #1 
Member Joined: Feb 2017 From: henderson Posts: 36 Thanks: 0  A bar of metal is cooling from explain! I want to understand on how to do this problem. Please try not to give me the answer thank you! 
April 12th, 2017, 01:21 AM  #2 
Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,155 Thanks: 731 Math Focus: Physics, mathematical modelling, numerical and computational solutions 
The formula at the end: $\displaystyle H = 23 + 977e^{0.1t}$ describes the temperature, $\displaystyle H$, at any given point in time, $\displaystyle t$. Note that $\displaystyle t$ is in minutes. It is an exponential decay function, so it will rapidly drop with increasing $\displaystyle t$ and then smooth out to a plateau. Try putting in different values of $\displaystyle t$ to calculate $\displaystyle H$ at different times. What value of $\displaystyle t$ do you need to substitute in to calculate $\displaystyle H$ after 1 hour? For the average part, you will need to use the formula for the average. Basically $\displaystyle \bar{x} = \frac{\int_{t_1}^{t_2} f(x) dx}{t_2t_1}$ This is very similar to the concept of "add up the numbers and divide by the number of numbers" except that your function is continuous, not discrete, so instead of adding numbers together, you perform an integral (which is sort of like a continuous sum) and instead of dividing by the number of numbers, you divide by the size of the interval that you want to calculate the average for. Therefore, if you wanted to calculate the average $\displaystyle H$ for the first hour $\displaystyle t_1 = 0$ $\displaystyle t_2 = 60$ $\displaystyle \bar{H} = \frac{\int_{0}^{60} (23 + 977e^{0.1t}) dt}{600}$ Let me know how it goes 

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