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4 years ago

Christine has nine shining rings. If six of them are gold, how many are not gold?

Mathematics

2 answers:

Anastasy [175]4 years ago

7 0

3 of them would not be good

9-6= 3 non shiny gold rings

Bess [88]4 years ago

5 0

The answer is bc 3 9-6=3

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Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint. (If an answer d

eduard

The Lagrangian is

$L(x_1,\ldots,x_n,\lambda_1,\ldots,\lambda_n)=x_1+\cdots+x_n+\lambda_1({x_1}^2+\cdots+{x_n}^2)+\cdots+\lambda_n({x_1}^2+\cdots+{x_n}^2)$

with partial derivatives (set equal to 0)

$\dfrac{\partial L}{\partial x_i}=1+2x_i(\lambda_1+\cdots+\lambda_n)=0$

$\dfrac{\partial L}{\partial\lambda_i}={x_1}^2+\cdots+{x_n}^2-36=0$

for each $1\le i\le n$ .

Let $\Lambda$ be the sum of all the multipliers $\lambda_i$ ,

$\Lambda=\displaystyle\sum_{k=1}^n\lambda_k=\lambda_1+\cdots+\lambda_n$

We notice that

$x_i\dfrac{\partial L}{\partial x_i}=x_i+2{x_i}^2\Lambda=0$

so that

$\displaystyle\sum_{i=1}^nx_i\dfrac{\partial L}{\partial x_i}=\sum_{i=1}^nx_i+2\Lambda\sum_{i=1}^n{x_i}^2=0$

We know that $\sum\limits_{i=1}^n{x_i}^2=36$ , so

$\displaystyle\sum_{i=1}^nx_i+2\Lambda\sum_{i=1}^n{x_i}^2=0\implies\sum_{i=1}^nx_i=-72\Lambda$

Solving the first $n$ equations for $x_i$ gives

$1+2\Lambda x_i=0\implies x_i=-\dfrac1{2\Lambda}$

and in particular

$\displaystyle\sum_{i=1}^nx_i=-\dfrac n{2\Lambda}$

It follows that

$-\dfrac n{2\Lambda}+72\Lambda=0\implies\Lambda^2=\dfrac n{144}\implies\Lambda=\pm\dfrac{\sqrt n}{12}$

which gives us

$x_i=-\dfrac1{2\left(\pm\frac{\sqrt n}{12}\right)}=\pm\dfrac6{\sqrt n}$

That is, we've found two critical points,

$\pm\left(\dfrac6{\sqrt n},\ldots,\dfrac6{\sqrt n}\right)$

At the critical point with positive signs, $f(x_1,\ldots,x_n)$ attains a maximum value of

$\displaystyle\sum_{i=1}^nx_i=\dfrac{6n}{\sqrt n}=6\sqrt n$

and at the other, a minimum value of

$\displaystyle\sum_{i=1}^nx_i=-\dfrac{6n}{\sqrt n}=-6\sqrt n$

4 0

4 years ago

Please help me ASAP

Murljashka [212]

Answer D and A

Step-by-step explanation:

8 0

3 years ago

Write the sum using summation notation, assuming the suggested pattern continues. -10 - 2 + 6 + 14 + ... + 110

Nimfa-mama [501]

This is the sum of the arithmetic sequence:
10 - 2 + 6 + 14 + ... + 110
where: a 1 = - 10, d = 8
a n = 110
a n = a 1 + ( n - 1 ) * d
110 = - 10 + ( n - 1 ) * 8
110 = - 10 + 8 n - 8
110 + 10 + 8 = 8 n
128 = 8 n
n = 128 : 8
n = 16
∑ n = n/2 * ( a 1 + a n )
∑ 16 = 16/2 * ( -10 + 110 ) = 8 * 100 = 800
Answer:
The sum is 800.

7 0

3 years ago

Read 2 more answers

Help?! Will give 20 points!

notka56 [123]

Do you need both or just 13?

8 0

3 years ago

What value is equivalent to 13 − 2(1 − 15) ÷ 4?

Bas_tet [7]

Use bodmas.
13-2(-14)/4
-11(-14)/4
154/4
46

7 0

3 years ago

Read 2 more answers