How do I solve this problem?
If the cooling system in a light-water nuclear reactor is shut off, the temperature of the fuel rods will increase. The temperature of the fuel rods during the first hour could be modeled by the equation T=680(1.0004)^t-665, where t is the time in seconds, and T is the temperature of the fuel rods in degrees Fahrenheit.
Average rate of change can be calculated by finding the slope between two points. Find the average rate at which the temperature changes for the first 30 minutes.
The given function is:
`T = 680(1.0004)^t - 665`
T represents the temperature of the fuel rods in degrees Fahrenheit and
t is the time in seconds
So to solve for the average rate of change function T for the first 30 minutes, we have to convert the given time to seconds.
`t = 30 m i n u tes * ( 60 seconds)/(1 m i n u te) = 180 seconds`
Since the time is first 30 minutes, then, the other value of t is 0 seconds.
Then, determine the corresponding values of T.
When t = 0,
`T =680(1.0004)^0 -665=680-665`
When t = 180
`T=680(1.0004)^180 -665 =730.7551137 - 665`
Then, apply the formula of average rate of change.
`a v e r a g e rate of chan g e = (f(b) - f(a))/(b-a)`
For our problem, it will be:
average of rate of change `= (T_2-T_1)/(t_2-t_1)`
average of rate of change `= (65.76-15)/(30-0)`
average of rate of change `= 1.692`
Therefore,the average rate at which the temperature changes for the first 30 minutes is 1.692 degree Fahrenheit per second.