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=1/10(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° and by the end of May the temperature was rising at the rate of 6° 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 dC/dt, the derivative of the crime rate with respect to time.
Given that T = 74° and dT/dt = 6° per month, we can substitute these values into the equation for the crime rate:
C = 1/10(74-60)^2 + 100
C = 1/10(14)^2 + 100
C = 1/10(196) + 100
C = 19.6 + 100
C = 119.6
Now, we need to find dC/dt by taking the derivative of the crime rate equation with respect to time:
dC/dt = 2/10(74-60)(dT/dt)
dC/dt = 2/10(14)(6)
dC/dt = 2/10(84)
dC/dt = 16.8
Therefore, the crime rate is rising at a rate of 16.8 crimes per month at the end of May.