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° 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 how fast 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.
Given: C = 1/10(T-60)^2 + 100
First, let's find the derivative of the crime rate function with respect to T:
dC/dT = 1/10 * 2(T-60) * 1
dC/dT = 1/5(T-60)
Now, we are given that the temperature is rising at a rate of 6° per month, so dT/dt = 6.
To find how fast the crime rate is rising at the end of May, we need to find dC/dt when T = 74 and dT/dt = 6:
dC/dt = (1/5)(74-60) * 6
dC/dt = (1/5)(14) * 6
dC/dt = 2.4
Therefore, the crime rate is rising at a rate of 2.4 crimes per month at the end of May.