An engineer, for example, may want to know how much work is required to pump all...

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An engineer, for example, may want to know how much work is required to pump all or part of the liquid from a container. To do this, we imagine lifting the liquid out one thin horizontal slab at a time and applying the equation W = Fd to each slab. We then evaluate the integral that this leads to as the slabs become thinner and more numerous. Suppose that a conical container has a height of 4 ft with a radius at the top of 2 ft and is filled to the top with freshwater. We may model the container in the plane with the coordinate axis intersected at the point (0, 0) (the origin) as illustrated below. Suppose that a slab has been created at an arbitrary value, y, with a thickness of Ay. Find the approximate volume of such a slab in terms of y. AV = ft³ Note: type 'Dy' to input Ay in your answer. Now suppose that the container is filled with freshwater having a weight-density of 62.4 and that the force required to lift this slab is equal to its weight. Find the force, F(y), required to lift this slab. F(y) = lb Hint: Weight = (weight per unit volume). volume. Note: Enter the function in terms of y and Ay. Type 'Dy' to input Ay in your answer. Setup the integral that will give the work done in pumping all of the water to the rim of the container. W = Note: Your answer should contain the differential, dy. Finally, compute the value of the work required to pump all of the liquid to the rim of the container. W= ft-lb An engineer, for example, may want to know how much work is required to pump all or part of the liquid from a container. To do this, we imagine lifting the liquid out one thin horizontal slab at a time and applying the equation W = Fd to each slab. We then evaluate the integral that this leads to as the slabs become thinner and more numerous. Suppose that a conical container has a height of 4 ft with a radius at the top of 2 ft and is filled to the top with freshwater. We may model the container in the plane with the coordinate axis intersected at the point (0, 0) (the origin) as illustrated below. Suppose that a slab has been created at an arbitrary value, y, with a thickness of Ay. Find the approximate volume of such a slab in terms of y. AV = ft³ Note: type 'Dy' to input Ay in your answer. Now suppose that the container is filled with freshwater having a weight-density of 62.4 and that the force required to lift this slab is equal to its weight. Find the force, F(y), required to lift this slab. F(y) = lb Hint: Weight = (weight per unit volume). volume. Note: Enter the function in terms of y and Ay. Type 'Dy' to input Ay in your answer. Setup the integral that will give the work done in pumping all of the water to the rim of the container. W = Note: Your answer should contain the differential, dy. Finally, compute the value of the work required to pump all of the liquid to the rim of the container. W= ft-lb

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