Suppose the percentage of drivers using seat belts from 2001 through 2009 is modeled by the function

Question

f(t) = 71.9(t + 1)^0.051

where t is measured in years, with t = 0

corresponding to the beginning of 2001. What was the average percentage use of this device over the period from the beginning of 2001 through the end of 2009 (t = 9)? (Round your answer to one decimal place.)

Answer 1

To find the average value of a function over a given integral, integrate the function over that interval, and divide by the length of the interval. That is find (1/(9-0)*S_[0,9]f(t)dt. (S in this case is meant to represent the integral sign.)

Answer 2

First, let's recall the definition of the average value of a function over an interval. Recalling the general definition of average, it's the sum of all the values in a given data set, divided by the number of values in the data set. For a function, it's the same thing: it's the sum of all values of the function over the given interval, divided by the length of the interval. In this case, our function is f(t) = 71.9(t+1)^0.051, and our interval is from 0 to 9.

The sum of all values of a function over a given interval, by definition, is the definite integral of that function over that interval. So to get the sum of all values of f(t) within the interval 0 to 9, we can calculate the integral from 0 to 9 of 71.9(t+1)^0.051 dt. Recall that you can factor constant terms out of the integral, so this becomes 71.9 times the integral from 0 to 9 of (t+1)^0.051 dt.

To solve this integral, we can apply u-substitution: u = t+1, and du = dt (since du/dt = 1). So our integral becomes u^0.051 du, and our limits of integration become 1 to 10 (remember, the original limits were in terms of t, so we have to recalculate them in terms of u using u = t+1). The integral for any function xⁿ where n is a constant is xⁿ⁺¹/(n+1), so we can solve this integral as u^1.051/1.051. Substituting in the (new) limits of integration, we get (10^1.051/1.051) - (1^1.051/1.051). (We can simplify this a bit, by factoring out the common denominator and remembering that 1 to the power of anything is 1: (10^1.051 - 1) / 1.051.)

Since we originally factored out 71.9 from the integral, we have to multiply it back again, so the value of the integral from 0 to 9 of 71.9(t+1)^0.051 dt is 71.9((10^1.051 - 1) / 1.051).

Finally, now that we have the value of the integral, we can divide this by the length of the original interval, which is 9 - 0 = 9. Our final answer, therefore, is (71.9((10^1.051 - 1) / 1.051)) / 9. Performing this final calculation on a calculator, we get that the average percentage of drivers using seat belts from 2001 to 2009 is 77.9 percent.