# geometry

posted by .

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 -

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.

## Similar Questions

1. ### algebra

A farmer decides to enclose a rectangular garden, using the side of a barn as one side of the rectangle. What is the maximum area that the farmer can enclose with 100 ft. of fence?
2. ### Algebra

You have 50 yards of fencing to enclose a rectangular region. Find the dimensions of the rectangle that maximize the enclosed area. What is the maximum area?
3. ### math

A farmer plans to enclose a rectangular region, using part of his barn for one side and fencing for the other three sides. If the side parallel to the barn is to be twice the length of an adjacent side, and the area of the region is …
4. ### pre-calc

area of a rectangular region: a farmer wishes to create two rectangular regions bordering a river, by three fences perpendicular to the river and one connecting them. suppose that x represents the length of each of the three parallel …
5. ### Algebra

A farmer decides to enclose a rectangular garden, using the side of a barn as one side of the rectangle. What is the maximum area that the farmer can enclose with 100 ft of fence?
6. ### algebra

A farmer decides to enclose a rectangular garden, using the side of a barn as one side of the rectangle. What is the maximum area that the farmer can enclose with 80 ft of fence?
7. ### Algebra

a farmer wants to enclose a rectangle garden using the sides of her barn as one side of the rectangle. what is the maximum area which she can enclose with 60ft of fence?
8. ### Algebra

A farmer plans to enclose a rectangular region using part of his barn for one side and fencing for remaining sides. The area of the region is maximized when the width is 30 feet. a.) write an equation that could model the area as a …
9. ### Functions

A farmer wants to enclose a rectangular field with 180m of fencing. The side of the barn will act as one side of the enclosure, leaving 3 sides to be covered. The function that describes the area inside is f(x)=x(180-2x) what are the …
10. ### Math(HELP!)

A farmer has 120 feet of fencing to enclose a rectangular plot for some of his animals. One side of the area borders on a barn. a.) If the farmer does not fence the side along the barn, find the length and width of the plot that will …

More Similar Questions

Post a New Question