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Mrac [35]
3 years ago
5

What are the solutions to the following system?

Mathematics
2 answers:
RoseWind [281]3 years ago
7 0
\bf \begin{cases}
-2x^2+y=-5\\
\boxed{y}=-3x^2+5\\
----------\\
-2x^2+\left( \boxed{-3x^2+5} \right)=-5
\end{cases}
\\\\\\
-5x^2+5=-5\implies -5(x^2-1)=-5\implies x^2-1=\cfrac{-5}{-5}
\\\\\\
x^2-1=1\implies x^2=2\implies x=\pm\sqrt{2}

and since we know what "x" is, then let's substitute it on say the 1st equation

\bf -2x^2+y=-5\implies -2\left( \pm \sqrt{2} \right)^2+y=-5\implies -4+y=-5
\\\\\\
y=-1\\\\
-------------------------------\\\\
\left( \sqrt{2}~,~-1 \right)\qquad \left( -\sqrt{2}~,~-1 \right)
NeX [460]3 years ago
6 0

Answer:

C

Step-by-step explanation:

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A piece of paper is to display ~128~ 128 space, 128, space square inches of text. If there are to be one-inch margins on both si
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Answer:

The dimensions of the smallest piece that can be used are: 10 by 20 and the area is 200 square inches

Step-by-step explanation:

We have that:

Area = 128

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Such that:

Length = x

Width = y

So:

Area = x * y

Substitute 128 for Area

128 = x * y

Make x the subject

x = \frac{128}{y}

When 1 inch margin is at top and bottom

The length becomes:

Length = x + 1 + 1

Length = x + 2

When 2 inch margin is at both sides

The width becomes:

Width = y + 2 + 2

Width = y + 4

The New Area (A) is then calculated as:

A = (x + 2) * (y + 4)

Substitute \frac{128}{y} for x

A = (\frac{128}{y} + 2) * (y + 4)

Open Brackets

A = 128 + \frac{512}{y} + 2y + 8

Collect Like Terms

A = \frac{512}{y} + 2y + 8+128

A = \frac{512}{y} + 2y + 136

A= 512y^{-1} + 2y + 136

To calculate the smallest possible value of y, we have to apply calculus.

Different A with respect to y

A' = -512y^{-2} + 2

Set

A' = 0

This gives:

0 = -512y^{-2} + 2

Collect Like Terms

512y^{-2} = 2

Multiply through by y^2

y^2 * 512y^{-2} = 2 * y^2

512 = 2y^2

Divide through by 2

256=y^2

Take square roots of both sides

\sqrt{256=y^2

16=y

y = 16

Recall that:

x = \frac{128}{y}

x = \frac{128}{16}

x = 8

Recall that the new dimensions are:

Length = x + 2

Width = y + 4

So:

Length = 8 + 2

Length = 10

Width = 16 + 4

Width = 20

To double-check;

Differentiate A'

A' = -512y^{-2} + 2

A" = -2 * -512y^{-3}

A" = 1024y^{-3}

A" = \frac{1024}{y^3}

The above value is:

A" = \frac{1024}{y^3} > 0

This means that the calculated values are at minimum.

<em>Hence, the dimensions of the smallest piece that can be used are: 10 by 20 and the area is 200 square inches</em>

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1. Given

2. AD is parallel to BC

3. ∠3 ≅ ∠2

4. transitive property of congruence

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