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nataly862011 [7]
3 years ago
15

Find three real numbers​ x, y, and z whose sum is 6 and the sum of whose squares is as small as possible. g

Mathematics
1 answer:
mart [117]3 years ago
8 0
We're minimizing x^2+y^2+z^2 subject to x+y+z=6. Using Lagrange multipliers, we have the Lagrangian

L(x,y,z,\lambda)=x^2+y^2+z^2+\lambda(x+y+z-6)

with partial derivatives

\begin{cases}L_x=2x+\lambda\\L_y=2y+\lambda\\L_z=2z+\lambda\\L_\lambda=x+y+z-6\end{cases}

Set each partial derivative equal to 0:

\begin{cases}2x+\lambda=0\\2y+\lambda=0\\2z+\lambda=0\\x+y+z=6\end{cases}

Subtracting the second equation from the first, we find

2x-2y=0\implies x=y

Similarly, we can determine that x=z and y=z by taking any two of the first three equations. So if x=y=z determines a critical point, then

x+y+z=3x=6\implies x=y=z=2

So the smallest value for the sum of squares is 2^2+2^2+2^2=12 when (x,y,z)=(2,2,2).
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THIS WILL BE 10 POINTS PER Q, SO IF YOU COULD HELP ANSWER THESE ALL, I WOULD LOVE YOU FOREVER :')
Contact [7]

1. C. 538.56

2. A. $1478.40

3. A. $2295.60

4.A.  $795.60

7 0
3 years ago
0.08 was rounded to 1 significant figure. What is the lower bound?
Black_prince [1.1K]
I believe the answer is 0.9 because a significant figure is a similar difference
7 0
3 years ago
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Draw a line representing the rise and a line representing the run of the line State the slope of the line in simplest form
amid [387]
Slope is -5/3
I hope this helps :)

8 0
3 years ago
The price of a sweatshirt at a local shop is twice the price of a pair of shorts. The price of a T-shirt at the shop is $4 less
fomenos

Answer:

The answer to your question is below

Step-by-step explanation:

Equation 1

                            $sweatshirt = 2 pair of shorts

                                    <u>  w = 2h</u>

Equation 2

$t-shirt = $pair of shorts  - $4

                                <u>      t = h - 4</u>

Equation 3

Total cost = 3 sweatshirts + 2 pairs of shorts + 5 t-shirts

<u>Total cost = 3w + 2h + 5t = 136 </u>    

5 0
3 years ago
Read 2 more answers
1)a chord of length 18cm midway the radius of a circle. calculate the radius of the circle correct to 1d.p. 2)if two parallel ch
kykrilka [37]
Part 1:

Given that the length of the chord is 18 cm and the chord is midway the radius of the circle. 

Thus, half the angle formed by the chord at the centre of the circle is given by:

\cos\theta=\frac{\left( \frac{1}{2} r\right)}{r}= \frac{1}{2}  \\  \\ \Rightarrow\theta=\cos^{-1}\left( \frac{1}{2} \right)=60^o

Now, 

\sin60^o= \frac{9}{r}  \\  \\ \Rightarrow r= \frac{9}{\sin60^o} =10.392

Therefore, the radius of the circle is 10.4 cm to 1 d.p.


Part 2I:

Given that the radius of the circle is 10 cm and the length of chord AB is 8 cm. Thus, half the length of the chord is 4cm. Let the distance of the mid-point O to /AB/ be x and half the angle formed by the chord at the centre of the circle be θ, then

\sin\theta= \frac{4}{10} = \frac{2}{5} \\ \\ \theta=\sin^{-1}\left( \frac{2}{5} \right)=23.6^o

Now, 

\cos23.6^o= \frac{x}{10} \\ \\ \Rightarrow x=10\cos23.6^o=9.165\approx9.2cm


Part 2II:

Given that the radius of the circle is 10cm and the angle distended is 80 degrees. Let half the length of chord CD be y, then:

\sin40^o= \frac{y}{10}  \\  \\  \\ \Rightarrow y=10\sin40^o=6.428

Thus, the length of chord CD = 2(6.428) = 12.856 which is approximately 12.9 cm.
3 0
3 years ago
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