The height ‘y’ of the rectangle is the distance from the top function to the bottom function. If we call the distance from x = 0 to one of the vertices ‘x’, the entire base can be written as ‘2x’. In this case, the rectangle is symmetrical on either side of the y-axis. To maximize the area, we must first find an equation describing the area. The rectangle whose area you want to maximize is shown: 2. The first step is to draw a diagram of the problem. Giant Rectangle of Doom! Optimization Problem.You can do this by implicit differentiation, and by using the chain rule: F’ = f’*g’(x) (If you consider x² to be a composite of functions, then the derivative of the inner function ‘x’ in relation to time is dx/dt, and the same is true for y and z.) Now you have an expression relating all the velocities, and all you have to do is plug in the values given in the problem to solve for the velocity of the UFO (dx/dt). The next step is to get an equation that relates the rates of change of the three distances. 6 ² + 2 ² = z ² z = 2 √10 km Negative, because it’s going towards the UFO, thus decreasing the distance. You can then use this to find the distance z. To solve this problem, you must first find an equation to relate x, y, and z.
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