Note that locating the locating the minimum point on the graph of S above can be done rigorously using calculus methods.The approximate value of x was found to be:.To find the x value that will minimize the surface area, we graph S as a function of x and locate the minimum point using a graphic calculator.Substitute h in the surface area formula by 3000 / x 2 to obtain a formula in terms of x only:.Solve the equation (1 / 3) h x 2 = 1000 for h to obtain:.In this problem we have L = W = x, hence: We now use the formula for the surface area found in problem 1 above to write a formula for the surface area S of the given pyramid.We first use the formula of the volume given above to write the equation:.All problems like the following lead eventually to an equation in that simple form. Yet, word problems fall into distinct types. Find the value of x so that the volume of the pyramid is 1000 cm 3 the surface area is minimum. W ORD PROBLEMS require practice in translating verbal language into algebraic language. Total Area = W * sqrt + L * sqrt + W * Līelow is shown a pyramid with square base, side x, and height h. Now the total area is obtained by adding the area of the base W * L to the area of the lateral surface.The total lateral area of the surface made up of the four triangles is given by:Ī(lateral surface) = W * sqrt + L * sqrt Triangle BOC is congruent to triangle AOD and have equal areas. Triangle AOB is congruent to triangle DOC and have equal areas.In a similar way as above, the area of triangle AOD is given by:Ī(AOD) = (1 / 2) * L * sqrt.Similarly, using triangle OSO" we can find a formula for the altitude H' of triangle AOD as follows:.We now use the length W of the base and the length H of the altitude of triangle DOC to find its area.OSO' is a right triangle and the length of the altitude H of triangle DOC is given by: (using Pythagora's theorem)
OS is orthogonal to the base ABCD of the pyramid and is therefore perpendicular to SO' and SO". Let O' be the middle of CD and O" be the middle of AD. Let S be the middle of the diagonal of the base ABCD of the pyramid. We need to find the area of each all these figures in order to find the area of the surface of the pyramid. The surface of the pyramid is made up of four triangles congruent in pairs and a rectangular base.Where L and W are the length and width of the base and h is the height of the pyramid.įind a formula for the total area of the surface of the pyramid shown above
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