The speed of a runner increased steadily during the first three seconds of a race. Her speed at half-second intervals is given in the table. Find lower and upper estimates for the distance that she traveled during these three seconds. ft (smaller value) ft (larger value)t (s) 0 0.5 1.0 1.5 2.0 2.5 3.0v (ft/s) 0 6.7 9.2 15.5 17.5 19.8 20.2

To find the lower and upper estimates for the distance that the runner traveled during these three seconds, we can use the concept of Riemann sums in calculus, which is a certain kind of approximation of an integral by a finite sum.

The given table provides the speed of the runner at half-second intervals. We can interpret these speeds as the average speeds over each half-second interval.

The lower estimate is found by taking the minimum speed during each interval, multiplying it by the time of the interval, and then summing up these products.

The upper estimate is found by taking the maximum speed during each interval, multiplying it by the time of the interval, and then summing up these products.

Lower Estimate: 0*(0.5-0) + 6.7*(1-0.5) + 9.2*(1.5-1) + 15.5*(2-1.5) + 17.5*(2.5-2) + 19.8*(3-2.5) = 0 + 3.35 + 4.6 + 7.75 + 8.75 + 9.9 = 34.35 ft

Upper Estimate: 6.7*(0.5-0) + 9.2*(1-0.5) + 15.5*(1.5-1) + 17.5*(2-1.5) + 19.8*(2.5-2) + 20.2*(3-2.5) = 3.35 + 4.6 + 7.75 + 8.75 + 9.9 + 10.1 = 44.45 ft

So, the lower and upper estimates for the distance that she traveled during these three seconds are 34.35 ft and 44.45 ft respectively.