Original Problem: Farmer Brown has 800 yards of fencing with which to build a rectangular corral, divided into two pens. He builds a corral that uses the river as one side, so he only has to fence the other 3 sides and the divider down the middle (perpendicular to the river).
I found the function of the Area to be A=x(800/3 - x/3), but I don't know how to find the maximum possible area of the rectangle. Can someone please explain this to me? Thanks.
How do I find the maximum area of the rectangle?
2017-02-15 7:38 am
回答 (3)
2017-02-15 8:16 am
You took the three lengths perpendicular to the river to be y and regarded the fencing parallel to the river to be a single length x, so that area A = xy
Substituting y from x + 3y = 800 your area function was
A=x(800 – x)/3 = (800/3)x – x^2/3
This is an inverted parabola so we expect a maximum area to exist
To find max area, differentiate and set that result to zero
dA/dx = 800/3 – (2/3)x = 0 at max
x = 400, y = 400/3, A(max) = 160,000/3
Substituting y from x + 3y = 800 your area function was
A=x(800 – x)/3 = (800/3)x – x^2/3
This is an inverted parabola so we expect a maximum area to exist
To find max area, differentiate and set that result to zero
dA/dx = 800/3 – (2/3)x = 0 at max
x = 400, y = 400/3, A(max) = 160,000/3
2017-02-15 8:10 am
x = side parallel to the river
y = side perpendicular to the river
800 = x + 3y
x = 800-3y
A = xy = (800-3y)y = 800y-3y²
A = -3y²+800y
parabola opens down so vertex is a maximum
vertex y = -b/(2a) = -800/-6 = 400/3 ft
x = 800-400 = 400 ft
A = xy = 160000/3 sq ft
= 53,333 1/3 sq ft
y = side perpendicular to the river
800 = x + 3y
x = 800-3y
A = xy = (800-3y)y = 800y-3y²
A = -3y²+800y
parabola opens down so vertex is a maximum
vertex y = -b/(2a) = -800/-6 = 400/3 ft
x = 800-400 = 400 ft
A = xy = 160000/3 sq ft
= 53,333 1/3 sq ft
2017-02-15 7:55 am
Area = x800/3 - x^2/3
derivative of area = 800/3 - 2x/3
Set that equal to 0, get 2x/3 = 800/3
x = 800/2 = 400.
In problems like this, it always seems to turn out that you want 1/2 your resources in one direction and 1/2 in the other. So you use 400 yards for the one side parallel to the river, and 400 divided into 3 pieces for the other sides perpendicular to the river.
So the area is 400 x (800-400)/3 = 160000/3 = 53333 1/3
derivative of area = 800/3 - 2x/3
Set that equal to 0, get 2x/3 = 800/3
x = 800/2 = 400.
In problems like this, it always seems to turn out that you want 1/2 your resources in one direction and 1/2 in the other. So you use 400 yards for the one side parallel to the river, and 400 divided into 3 pieces for the other sides perpendicular to the river.
So the area is 400 x (800-400)/3 = 160000/3 = 53333 1/3
收錄日期: 2021-04-24 00:17:24
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