Posted by steph on Thursday, November 10, 2011 at 8:28pm.
construction of a rain gutter: a piece of rectangular sheet metal is 20 inches wide. it is to be made into a rain gutter by turning the edges to form parallel sides. let x represent the length of each of the parallel.
a) give the restrictions on x.
b) determine a function of A that gives the area of a cross section of the gutter.
c) for what values of x will A be a maximum (and thus maximize the amount of the water that the gutter will hold)?
d) what is this maximum area?
e) for what values of x will the area of a cross section be less than 40 square inches?
- pre-calc - Reiny, Thursday, November 10, 2011 at 9:38pm
Look at a cross-section of the gutter.
let the height be x
then the base width = 20-2x , assuming the gutter is open at the top
a) in 20-2x , 0 < x < 10
b) Area = x(20-2x)
c) Area = 20x - 2x^2
= -2(x^2 - 10x + 25 - 25 )
= -2( (x-5)^2 - 25)
= -2(x-5)^2 + 50
d) maximum area is 50 when x = 5
e) x(20-2x) < 40
-2x^2 + 20x < 40
x^2 - 10x + 20 > 0
consider x^2 - 10x + 20 = 0
x = (10 ± √20)/2
= 5 ± √5
so the area is less than 40 for
5-√5 < x < 5+√5
- pre-calc - steph, Thursday, November 10, 2011 at 9:47pm
thank you! that helped a lot
- pre-calc - jerry, Monday, September 19, 2016 at 6:24pm
how did you figure c?
Answer This Question
More Related Questions
- Pre-Calc) - 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 ...
- calculus - A long rectangular sheet of metal, 12 inches wide, is to be made into...
- Calculus - A long rectangle sheet of metal, 15 inches wide is to be made into a ...
- math - You are designing a rain gutter made from a piece of sheet metal that is...
- math problem - A simple rain gutter is constructed using a sheet of aluminum ...
- mathematics - Please help me answer this question; 1.A metal gutter open at the ...
- Math Wrod Problem - 1. A long strip of copper 8 inches wide is to be made into a...
- Algebra 2 - A rain gutter is made by bending up the edges of a piece of metal of...
- Calculus - A 10-foot section of gutter is made from a 12-inch-wide strip of ...