AS maths help ........ please[quadratics questions]?

Question

a farmer has 60m of fenceing to make a pen for a goat. he wants to make the area of the pen as large as possible.
The length of the pen is x meters
1)write down an expresision in terms of x for the width of the pen
2) find an expression for the area of the pen
3) write this expression in the completed square form
4) what value of x make the area af the pen as large as possible. What is this area?

Answer 1

Part 1
Let sides be x and y
x + y = 30
y = 30 - x

Part 2
A(x) = x ( 30 - x )

Part 3
A(x) = 30 x - x ²
A(x) = - (x ² - 30 x)
A(x) = - ( x ² - 30 x + 225 - 225 )
A(x) = - [ (x - 15) ² - 225 ]

Part 4
x = 15 for largest value of A(x)
A = 225 m ² is largest area of pen.

Answer 2

60m of fencing
x = length
area = length x width x (30-x) = 30x -3x^2
1/2 (60 - 2x) = w
(30 - x) = w

x (30-x) =
30x -3x^2 =
1st derivative is 30-6x = 0

x = 5 w = 25 max area = 225 sq m

this many requests you should pay money
1)write down an expresision in terms of x for the width of the pen
2) find an expression for the area of the pen
3) write this expression in the completed square form
4) what value of x make the area af the pen as large as possible. What is this area?

Answer 3

1. The width is 30-x
(60-2x/2)
2. The area A=x*(30-x)
3. A=-x^2+30x
4. The value that maximizes the
area A stands for x=-30/-2=15
Then the area is a square and
finally A=15^2=225

Answer 4

The pen must be a regular four sided figure since it has a length and width.

1) If the length is x. And fencing is 60m, then 2 times the length + 2 times the width=60m. (That is perimeter of the four sided figure is 60m).

So;
2width +2x=60
2width=60-2x
width=30-x

2) Area= length* width (length multiplied by width)

Therefore area=x(30-x)

3) Area=30x-x^2

4) I don't know what level of math you're doing but I'll assume you know what differentiation is, and you know what maxima and minima are. Using this principle,

Area(a)=30x-x^2
da/dx (differential of area with respect to x)= 30-2x

Area(a) is maximum when da/dx=0

Therefore for a to be maximum, 30-2x=0
-2x=-30
x=15

The are therefore will be 30*15-15^2
=450-225
=225

Answer 5

1. width expression:
2(x + x) = 60
2(2x) = 60
4x = 60
x = 15

2. area of the pen:
= x^2

3. completed square form:
= 15^2

4. area:
= 225 square meters

Answer 6

Truman B is right - te maximum area enclosed by any given length of fencing is circular. however the question DID mention a width so we must assume the questioner wishes us to consider a rectangular pen. (It would get more complicated if we had a genarl quadrilateral.) Assuming a rectangular pen, the above answers are correct (although there are mistakes in one of two of them!)

Answer 7

are you sure that the square would be greater than a circular pen? 268.5 sq/m vs a square with 225