Question

A sheet of metal is 30cm x 6m and is being used to make a rectangular eavestrough by bending along the dotted lines. What value of x maximizes the capacity of the eavestrough?

Answer 1

instead of x im going to use the letter b, so it isnt confused with the times symbol the volume is Length x height x width = Volume (6) x (b) x (0.3 - 2b) = V 1.8b - 12b^2 = V the first derivative of V with respect to b is dV/db = 1.8 - 24b this is at a maximum (or possibly minimum) when its first derivative is equal to zero 0 = 1.8 - 24b solving for b b = 1.8/24 = 0.075m or 7.5 cm