STATEMENT OF PROBLEM

When a batter comes to the plate in baseball and swings at a pitch, the swing usually takes 0.2 seconds to reach the plate. During this time, the batter is transferring energy to the bat. The rate at which the energy is transferred to the bat increases from 0 to 9 horsepower during the first 0.15 seconds and then decreases to zero as the bat comes across the plate. (Note: one horsepower equals 746 joules/second.) The following data was collected for a particular swing:

Time Power
0 0.0
.025 0.8
.05 2.0
.075 3.0
.1 3.0
.125 7.5
.15 9.0
.175 3.0
.2 0.0

1. Connect the data points in a piecewise linear graph on a Cartesian coordinate system with power (in horsepower) versus time (in seconds).

2. Using your graph, predict the time interval in which the power is increasing the fastest.

3. Use the trapezoid rule to estimate the area under the graph which represents the total energy transferred to the bat in a given time interval. Be certain to label the units of measure for total energy transferred.

4. Find a polynomial function which fits this data.

5. Based on your polynomial, at what instant is the rate of energy transfer increasing the fastest?

6. Based on your polynomial, find a new function which represents the cumulative energy transfer as a function of time and plot the graph. Note: if you plot ten times the cumulative energy transfer, you can plot both graphs on the same set of axes (understanding that the vertical scale represents different quantities.)