Sociologists have found that crime rates are influenced by temperature. In a town of 150,000 people the crime rate has been approximated as C=1/5(T-60)^2+110, 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 73° and by the end of May the temperature was rising at the rate of 8° per month. How fast is the crime rate rising at the end of may?
To find how fast the crime rate is rising at the end of May, we need to find dC/dt when T=73° and dT/dt=8°.
Given that C=1/5(T-60)^2+110, we can first find dC/dT by taking the derivative of C with respect to T:
dC/dT = 2/5(T-60)
Next, we substitute T=73 and dT/dt=8 into dC/dT to find the rate at which the crime rate is increasing:
dC/dt = (2/5)(73-60)*8
dC/dt = 2/5 * 13 * 8
dC/dt = 2/5 * 104
dC/dt = 208/5
dC/dt = 41.6
Therefore, the crime rate is rising at a rate of approximately 41.6 crimes per month at the end of May.