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?

Question

campbell_st · Answer

draw a picture

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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