A strip of sheet metal 30 cm wide is to be made into a trough by turning strips up vertically along two sides. How many centimeters should be turned up at east side to obtain the greatest carrying capacity?

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Answer 1 · 2016-03-14T23:34:00

#7.5#cm

Let the 'turn up' be #t# #"cm"#.

Then twice the cross sectional area in #"cm"^2# is:

#2t*(30-2t) = 60t-4t^2 = 15^2-(4t^2-60t+15^2) = 225-(2t-15)^2#

Now #(2t-15)^2 >= 0# for any Real value of #t#. Its minimum possible value #0# is obtained when #(2t-15) = 0#. That is when #t = 7.5#.

This minimum value of #(2t-15)^2# results in the maximum value of #225-(2t-15)^2#.

Here's a graph of #1/10# cross sectional area against 'turn up':
graph{(225-(2x-15)^2)/20 [-14.83, 25.17, -5.92, 14.08]}