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A rectangular picture frame with total area 50000 cm2 includes a border which is 1 cm thick at the top and the bottom and 5 cm thick at the left and right side. What is the largest possible area of a picture that can be displayed in this frame?
draw a picture |dw:1371797342461:dw| you know the area of the picture + frame is 5000 so \[5000 = l \times w\] make w the subject so \[w = \frac{l}{5000}\] and you know the area of the pictues is \[A_{p} = ( l - 10)(w - 2)\] substitute w \[A_{p} = (l - 10)(\frac{l}{5000} - 2)\] you now have an area equation in one unknown find the 1st derivative. set it to 0 and solve for l. this should given the necessary dimensions