A hydraulic cylinder on an industrial machine pushes a steel block a distance of x feet $( 0 \leq x \leq 4 )$, where the variable force required is $F ( x ) = 3000 x e ^ { - x }$ pounds. Find the work done in pushing the block the full 4 feet through the machine. Round your answer to three decimal places.
A) $2725.265 \mathrm { ft } . \mathrm { lbs }$
B) $2890.106 \mathrm { ft } . \mathrm { lbs }$
C) $1655.749 \mathrm { ft } . \mathrm { lbs }$
D) $2839.235 \mathrm { ft } . \mathrm { lbs }$
E) $3219.788 \mathrm { ft }$.lbs

Question

Quizplus · Accepted Answer

The work done by a variable force is calculated by integrating the force function over the distance moved. For the given force function $F(x) = 3000xe^{-x}$, the work done over a distance of 4 feet is found by integrating $F(x)$ from 0 to 4. The integral of $3000xe^{-x}$ from 0 to 4 yields approximately 2725.265 ft.lbs, making choice A correct.

A Hydraulic Cylinder on an Industrial Machine Pushes a Steel (0≤x≤4)( 0 \leq x \leq 4 )(0≤x≤4)

A Hydraulic Cylinder on an Industrial Machine Pushes a Steel $( 0 \leq x \leq 4 )$