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SIZIF [17.4K]
3 years ago
9

Alex has 520 yards of fencing to enclose a rectangular area.

Mathematics
2 answers:
Pavlova-9 [17]3 years ago
6 0
<span>Perimeter =2w+2L= 520.
We can solve this by understanding that the area is maximized by a square
Therefore L=w.

p=2w+2w=520=4w
w=130

Area
 
A=wL=130(130)= 16900 square yards</span>
Kobotan [32]3 years ago
6 0

The first thing we are going to do for this case is define variables.

We have then:

w: width

l: length

The perimeter is given by:

2w + 2l = 520

The area is given by:

A = w * l

The area as a function of a variable is:

A (w) = w * (260-w)

Rewriting we have:

A (w) = -w ^ 2 + 260w

To obtain the maximum area, we derive:

A '(w) = -2w + 260

We equal zero and clear the value of w:

-2w + 260 = 0\\2w = 260

w = \frac{260}{2}\\w = 130

Then, the length is given by:

l = 260 - w\\l = 260 - 130\\l = 130

Finally, the maximum area obtained is:

A = w * l\\A = 130 * 130\\A = 16900

Answer:

A retangle that maximizes the enclosed area has a length of 130 yards and a width of 130 yards.

The maxium area is 16900 square yards

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7 0
3 years ago
Numbers that add to -0.5 and multiply to 1
Oliga [24]
Short Answer
There are two numbers
x1 = -0.25 + 0.9682i   <<<< answer 1
x2 = - 0.25 - 0.9582i   <<<< answer 2 
 
I take it there are two such numbers.
Let one number = x
Let one number = y

x + y = -0.5    
y = - 0.5 - x       (1)
xy = 1               (2)

Put equation 1 into equation 2
xy = 1
x(-0.5 - x) = 1
-0.5x - x^2 = 1               Subtract 1 from both sides.
-0.5x - x^2 - 1 = 0          Order these by powers
-x^2 - 0.5x -1 = 0           Multiply though by - 1
x^2 + 0.5x + 1 = 0         Use the quadratic formula to solve this.

\text{x = }\dfrac{ -b \pm \sqrt{b^{2} - 4ac } }{2a}

a = 1
b = 0.5
c = 1


\text{x = }\dfrac{ -0.5 \pm \sqrt{0.5^{2} - 4*1*1 } }{2*1}

x = [-0.5 +/- sqrt(0.25 - 4)] / 2
x = [-0.5 +/- sqrt(-3.75)] / 2
x = [-0.25 +/- 0.9682i

x1 = -0.25 + 0.9682 i
x2 = -0.25 - 0.9682 i

These two are conjugates. They will add  as x1 + x2 = -0.25 - 0.25 = - 0.50.

The complex parts cancel out.  Getting them to multiply to 1 will be a little more difficult. I'll do that under the check.

Check
(-0.25 - 0.9682i)(-0.25 + 0.9682i)
Use FOIL
F:-0.25 * -0.25 = 0.0625
O: -0.25*0.9682i
I: +0.25*0.9682i
L: -0.9682i*0.9682i = - 0.9375 i^2 = 0.9375

Notice
The two middle terms (labled "O" and "I" ) cancel out. They are of opposite signs.

The final result is 0.9375 and 0.0625 add up to 1
 
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Answer:

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