Sociologists have found that crime rates are influenced by temperature. In a town of 200,000 people, the crime rate has been approximated as C=1/10(T-60)^2+100, where C is the number of crimes per month and T is the average monthly temperature in degrees Fahrenheit. The average temperature for May was 74 degrees and by the end of May the temperature was rising at the rate of 6 degrees per month. How fast is the crime rate rising at the end of May?
To find the rate at which the crime rate is rising at the end of May, we need to find the derivative of the crime rate function with respect to temperature, and then plug in the values for the temperature and its rate of change.
Given: C = 1/10(T-60)^2 + 100
Taking the derivative of C with respect to T:
dC/dT = 1/10 * 2(T-60) * 1 + 0
= 1/5(T-60)
Now, we can substitute T = 74 and dT/dt = 6 into the equation:
dC/dt = 1/5(74-60) * 6
= 1/5(14) * 6
= 84/5
= 16.8
Therefore, the crime rate is rising at a rate of 16.8 crimes per month at the end of May.