Posted by **david** on Tuesday, December 13, 2011 at 5:31pm.

A farmer has 30yd of fencing and wants to enclose an area beside his barn. What are the dimensions of the region of maximal area, and what is that maximal area if the region is an is isosceles triangle with its base along side of the barn. Given -> legs of length 15, x = 1/2 base

- geometry -
**Reiny**, Tuesday, December 13, 2011 at 5:57pm
let the base be 2x ,(if x = 1/2 the base)

and let each of the equal sides be y

so we know 2x + 2y = 30

x+y = 15, or y = 15-x

let the height of the triangle be h

then h^2 + x2 = y^2

h^2 + x^2 = (15-x)^2

h^2 + x^2 = 225 - 30x + x^2

h = √(225-30x)

area = (1/2)base x height

= (1/2)(2x)(√(225-30x) )

= x (225-30x)^(1/2)

d(area) = x(1/2)(225-30x)^(-1/2) (-30) + (225-30x)^(1/2)

=0 for a max of area

(225-30x)^(1/2) = 15x/(225-30x)^(1/2)

cross-multiply

15x = 225-30x

45x = 225

x = 5

then h = √(225-150) = √75

and since x+y = 15

y = 10

so each side of the triangle must be 10 yds

and the largest area is (1/2)(10)(√75)

= 25√3

Makes sense that the triangle of largest area would be an equilateral triangle, just like the largest quad would be a square, etc.

## Answer This Question

## Related Questions

- Algebra - A farmer plans to enclose a rectangular region using part of his barn ...
- math - A farmer plans to enclose a rectangular region, using part of his barn ...
- algebra - A farmer decides to enclose a rectangular garden, using the side of a ...
- algebra - A farmer decides to enclose a rectangular garden, using the side of a ...
- Algebra - A farmer decides to enclose a rectangular garden, using the side of a ...
- pre-calc - area of a rectangular region: a farmer wishes to create two ...
- Algebra - a farmer wants to enclose a rectangle garden using the sides of her ...
- Math(HELP!) - A farmer has 120 feet of fencing to enclose a rectangular plot for...
- Algebra - You have 50 yards of fencing to enclose a rectangular region. Find the...
- Functions - A farmer wants to enclose a rectangular field with 180m of fencing. ...

More Related Questions