How to find the x and y coordinates of an extrema ? Take the first derivative of the function set equal to zero and solve for x. That give a A(r) min (minimum surface area). To do so we will have to minimize the surface area of the tank and to calculate the radius r and height h We must use the smallest possible quantity of metal. We want to get the cheapest possible tank with a volume of 64 m 3. Isolate h in equation (1): h = 64 / (π r 2) (2) Surface area : A(r) = Lid area + Side area = 2 π r 2 + 2 π r h (1) Volume of the cylinder: Vol = π r 2 h = 64 m 3 The surface area of the cylinder is symbolized by A(r), which is a function of radius r: To find the surface area of the cylinder you need to add the surface area of both end (the lids) and the surface area of the side. Let us draw a schema of the cylindrical tank Important note: the cylindrical tank is closed at both ends by a lid. The point is to save raw material and thus Earth's limited resources. You are asked to find the dimensions of that cylindrical tank in order to minimize the quantity of steel required to make it. Therefore a huge steel cylindrical tank contains 64 m 3 of paint (when it is full). However, before the car is assembled, the doors of the car must be painted. Nevertheless he invented and created the first moving assembly line in 1906). In a factory where cars are being manufactured on an Assembly Line, the car is assembled thanks to the genius of the American founder of the Ford Motor Company (Henry Ford was openly sympathetic to the Nazis and Anti-Semite. Mathematics and derivatives : Solution exercise 1.5 ( Advanced placement math level)ġ.5 Optimization : Henry Ford and its moving assembly line Mathematical tools to use: first derivative and second derivative, methods for solving a system of linear equations,įormula to compute a disc area and formula for finding the volume of a cylinder. The dimensions that save a maximum of raw material. Optimization exercise for advanced placement calculus classes (AP Calculus) : minimize the surface area of a cylindrical tank made of steel (with a given volume) in order to get
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