The change from religious to lay teachers at Roman Catholic schools has been partly attributed to the decline in the number of women and men entering religious orders.The percent of teachers who are lay teachers is given by where t is measured in decades, with corresponding to the beginning of 1960. Find the year when the percent of lay teachers was increasing most rapidly.
A)1970
B)1983
C)1976
D)2000
E)1962

Question

The change from religious to lay teachers at Roman Catholic schools has been partly attributed to the decline in the number of women and men entering religious orders.The percent of teachers who are lay teachers is given by  where t is measured in decades, with  corresponding to the beginning of 1960. Find the year when the percent of lay teachers was increasing most rapidly.
A)1970 
B)1983 
C)1976 
D)2000 
E)1962

Quizplus · Accepted Answer

To find when the percent of lay teachers was increasing most rapidly, we need to find the maximum of the first derivative of the given function. Taking the derivative with respect to t, we get:

y'(t) = 11eb12b0_a99e_7402_b394_1b0fd8ecbe68_TB7866_11' = 0.11e^(0.12t)

Setting y'(t) = 0, we get t = ln(0) / 0.12, which is undefined. This means that the function never decreases or reaches a maximum. However, we can find the year when the function was increasing most rapidly by finding the maximum of the second derivative:

y''(t) = (11e^(0.12t))' = 1.32e^(0.12t)

Setting y''(t) = 0, we get t = ln(0) / 0.12, which is again undefined. This means that the second derivative is always positive, so the function is always concave up and increasing most rapidly at the latest time period. Therefore, we choose the latest time period, which corresponds to the end of 1990, or 1983. Therefore, the answer is B.

The Change from Religious to Lay Teachers at Roman Catholic