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Story Problem: A gardening company needs to fill a large cylindrical planter with soil for a city park. The planter has a diameter of 4 feet and a height of 3 feet. However, at the center of this cylindrical planter, there is a decorative cone that is not to be filled with soil. The cone has a diameter of 2 feet and a height of 2.5 feet. Calculate the volume of soil needed to fill the cylindrical planter around the cone, using the formulas for the volume of a cylinder and a cone. Solution: First, we find the volume of the cylinder, the formula for which is V = πr²h. The radius r of the cylinder is half of its diameter, so r = 4 feet / 2 = 2 feet. Hence, the volume of the cylinder V_cylinder is: V_cylinder = π(2 feet)²(3 feet) V_cylinder = π(4 square feet)(3 feet) V_cylinder = 12π cubic feet Next, we calculate the volume of the cone using the formula V = (1/3)πr²h. The radius of the cone r is half of its diameter, so r = 2 feet / 2 = 1 foot. The volume of the cone V_cone is: V_cone = (1/3)π(1 foot)²(2.5 feet) V_cone = (1/3)π(1 square foot)(2.5 feet) V_cone = (2.5/3)π cubic feet V_cone = (5/6)π cubic feet To find the volume of soil needed, we subtract the volume of the cone from the volume of the cylinder: Volume of soil = V_cylinder - V_cone Volume of soil = 12π cubic feet - (5/6)π cubic feet Volume of soil = (72/6)π cubic feet - (5/6)π cubic feet Volume of soil = (67/6)π cubic feet Therefore, the company needs (67/6)π cubic feet or approximately 35.1 cubic feet of soil to fill the planter around the decorative cone.

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