Posted by MEG on Monday, November 15, 2010 at 9:06pm.
A long rectangular sheet of metal, 12 inches wide, is to be made into a rain gutter by turning up two sides so that they are perpendicular to the sheet. How many inches should be turned up to give the gutter its greatest capacity?

calculus  Reiny, Monday, November 15, 2010 at 9:44pm
All we have to consider is the area of the crosssection, since the length is a constant.
let the sides to be turned up at both ends be x inches long
let the remaining base be y inches
2x + y = 12
y = 122x
area = xy = x(122x) = 12x  2x^2
d(area)/dx = 12  4x = 0 for a max of area
4x = 12
x = 3
So the turnups should be 3 inches

calculus  profkn, Tuesday, November 16, 2010 at 1:23am
Greatest capacity will be achieved when the area of cross section will be maximum.
Let X be the base ,so (12X)/2 will be the height.
Area= Base x height= X(12X)/2
=(12XX^2)/2
Following the maximum theory, we differentiate the area w.r.t X and equate it to zero.
d/dX of(Area)=(122X)/2 =0
or 6X=0
or X=6 inches(base)
so, height= (126)/2 = 3 inches
Answer This Question
Related Questions
 Calculus  A long rectangle sheet of metal, 15 inches wide is to be made into a ...
 precalc  construction of a rain gutter: a piece of rectangular sheet metal is ...
 PreCalc)  a piece of rectangular sheet metal is 20 in wide. It's to be made ...
 geometry  a piece of rectangular sheet metal is 20 in. wide. It is made into a ...
 Math Wrod Problem  1. A long strip of copper 8 inches wide is to be made into a...
 math  You are designing a rain gutter made from a piece of sheet metal that is...
 Calculus  A carpenter wants to make an opentopped box out of a rectangular ...
 Algebra  A box is to be formed from a rectangular piece of sheet metal by ...
 spc  a long strip of sheet metal 12 inches wide is to be formed into an open ...
 math  A 160‐inch strip of metal 20 inches wide is to be made into a small...
More Related Questions