Suppose the sand in a sand box with length 14 feet, width 8 feet, and height 1.5 feet is to be removed. The density of the sand (in pounds p

Question

Suppose the sand in a sand box with length 14 feet, width 8 feet, and height 1.5 feet is to be removed. The density of the sand (in pounds per cubic feet) h feet from the bottom is given by δ(h)=2.5−h. Find the work required to completely empty the sandbox.

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Huyền Thanh 4 years 2021-08-02T16:47:57+00:00 1 Answers 42 views 0

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  1. lethu
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    2021-08-02T16:49:41+00:00
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    Answer:

    The work required is  W= 252lb \cdot ft

    Explanation:

      The volume of the sand box is mathematical represented as

                         dV = L * W * dh

    Substituting 14 feet for L,  8 feet for W into the equation

                               = 14* 8*dh

                              = 112dh

    The force as a result of the sand in the disk is mathematically represented as

                  dF = \rho * dV

         Substituting (2.5-h)  for \rho

                  dF =112(2.5-h) dh

    Now the work that is required to lift the sand from h = 0 to a height of  h=1.5 m is mathematically represented as

                          dW = 112(2.5 -h)(1.5-h)dh

    Now above is the formula for change in work done in order to obtain the workdone we integrate

                        W = 112 \int\limits^{1.5}_0 {(2.5 - h)(1.5-h)} \, dh

                             =112\int\limits^{1.5}_0 {3.5-4h + h^2} \, dh  

                            = 112 [3.75h -2h^2 + \frac{h^3}{3} ]{ {{1.5} \atop {0}} \right.

                           = 112 [3.75 (1.5) -2(1.5)^2 + \frac{1.5^3}{3} ]

                          = 112 * 2.25

                         = 252lb \cdot ft

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