Chapter 7.5, Problem 49E

Modeling Data The hydraulic cylinder on a woodsplitter has a 4-inch bore (diameter) and a stroke of 2 feet. The hydraulic pump creates a maximum pressure of 2000 pounds per square inch. Therefore, the maximum force created by the cylinder is $2000\left(\pi \cdot 2^2\right)=8000\pi$ pounds.

(a) Find the work done through one extension of the cylinder, given that the maximum force is required.

(b) The force exerted in splitting a piece of wood is variable. Measurements of the force obtained in splitting a piece of wood are shown in the table. The variable x measures the extension of the cylinder in feet, and F is the force in pounds. Use the regression capabilities of a graphing utility to find a fourth-degree polynomial model for the data. Plot the data and graph the model.

x	0	$\frac{1}{3}$	$\frac{2}{3}$	1	$\frac{4}{3}$	$\frac{5}{3}$	2
F(x)	0	20,000	22,000	15,000	10,000	5000	0

(c) Use the model in part (b) to approximate the extension of the cylinder when the force is maximum.

(d) Use the model in part (b) to approximate the work done in splitting the piece of wood.