1.Find the point P on the line y=3x that is closest to the point (60,0). What is the least distance between P and (60,0). 2. A silo (base not included) is to be constructed in the form of a cylinder surmounted by a hemisphere. The cost of construction per square unit of surface area is 10 times as great for the hemisphere as it is for the cylindrical sidewall. Determine the dimensions to be used if the volume is fixed at 2000 cubic units and the cost of construction is to be kept to a minimum. Neglect the thickness of the silo and waste in construction. ---Find the radius of the cylindrical base (and of the hemisphere)---Find the height
Any point of the line y = 3x is (x, 3x). The distance from (60,0) to (x,3x) is: D(x) = √[(x-60)? + (3x-0)?] D(x) = √[10x? - 120x + 3600] = √[10(x? - 12x + 360)] If you're in calculus, you can take the derivative of D(x) and find the point that's the relative minimum. If you're not in calculus, then you can find the vertex of the quadratic expression under the square root. The x-coordinate of the vertex is -b/(2a) = 120/20 = 6. Then the point is (6, 18), and the minimum distance is 18√10. _______________________________________ The volume of the silo is the sum of the cylinder's volume and hemisphere's volume, and the volume equals 2000 units?. V = πr?h + ?πr? 2000 = πr?h + ?πr? (2000 - ?πr?)/πr? = h = 2000/(πr?) - 2r/3 The cost of the construction will include the lateral area of the cylinder, 2πrh, and the area of the hemisphere, 2πr?. To reflect the fact that the hemisphere will cost 10 times as much, we multiply its surface area by 10: C = 2πrh + 10(2πr?) C = 2πrh + 20πr? Substitute h = 2000/(πr?) - 2r/3 into the Cost equation: C = 2πr·(2000/(πr?) - 2r/3) + 20πr? C = 4000r?? - 4πr?/3 + 20πr? C = 4000r?? + 56πr?/3 Again, calculus vs. no calculus. Without calculus, you can find the minimum using a graphing calculator since it's not a quadratic in this case. With calculus: C' = -4000r?? + 112πr/3 0 = -4000r?? + 112πr/3 4000/r? = 112πr/3 12000/(112π) = r? = 750/(7π) r = 5?(6/(7π)) ≈ 3.24 units h = 2000/(π3.24?) - 2(3.24)/3 ≈ 58.45 units
