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Talja [164]
2 years ago
10

Find the point (x,y) of x2+14xy+49y2=100 that is closest to the origin and lies in the first quadrant.

Mathematics
1 answer:
Gala2k [10]2 years ago
3 0

Notice that

x^2+14xy+49y^2=(x+7y)^2

so the constraint is a set of two lines,

(x+7y)^2=100\implies\begin{cases}x+7y=10\\x+7y=10\end{cases}

and only the first line passes through the first quadrant.

The distance between any point (x,y) in the plane is \sqrt{x^2+y^2}, but we know that \sqrt{f(x,y)} and f(x,y) share the same critical points, so we need only worry about minimizing x^2+y^2. The Lagrangian for this problem is then

L(x,y,\lambda)=x^2+y^2+\lambda(x+7y-10)

with partial derivatives (set equal to 0)

L_x=2x+\lambda=0

L_y=2y+7\lambda=0

L_\lambda=x+7y-10=0

We have

L_y-7L_x=2y-14x=0\implies y=7x

which tells us that

x+7y-10=0\iff x+49x=10\implies x=\dfrac15\implies y=\dfrac75

so that \left(\dfrac15,\dfrac75\right) is a critical point. The Hessian for the target function x^2+y^2 is

H(x,y)=\begin{bmatrix}2&0\\0&2\end{bmatrix}

which is positive definite for all x,y, so the critical point is the site of a minimum. The minimum distance itself (which we don't seem to care about for this problem, but we might as well state it) is \sqrt{\left(\dfrac15\right)^2+\left(\dfrac75\right)^2}=2.

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An oil exploration company currently has two active projects, one in Asia and the other in Europe. Let A be the event that the A
Ksivusya [100]

Answer:

0.7 ; 0.65 ; 0.115

Step-by-step explanation:

Step-by-step explanation:

P(A) = 0.5 ; P(B) = 0.3

P(not successful) = P(B') = 1 - 0.3 = 0.7 ; P(A') = 1 - 0.5 = 0.5

1.)

Both events are independent events, hence the outcome of one does not depend on the other. That is the failure of the Asian project has nothing to do with the European project.

Probability that European project isn't successful;

P(B') = 1 - P(B) = 1 - 0.3 = 0.7

2.)

Probability that atleast one of the 2 projects is successful :

P(AUB) = P(A) + P(B) - P(AnB)

P(AnB) = P(A) * P(B) = 0.5 * 0.3 = 0.15

P(AUB) = 0.5 + 0.3 - 0.15 = 0.65

3.)

Probability that only the Asian project is successful, given that atleast one of the two projects is successful :

[P(A) - P(AnB)] ÷ P(AuB)

[0.5 * 0.15] ÷ 0.65

= 0.075 ÷ 0.65

= 0.1153846

= 0.115

6 0
1 year ago
Help plz
Ber [7]

Answer:

Definition of bisector

Angle addition postulate

same side interior angles theorem

subtraction property of equality

Step-by-step explanation:

It is given that m║n, m∠1 = 65°,m∠2=60° and BD --> bisects ∠ABC . Because of triangle sum theorem, m∠3 = 55°. By the definition of bisector , ∠3 ≅∠4, so m∠4= 55°. Using the angle addition postulate , m∠ABC = 110°, m∠5 = 110° because vertical angles are congruent . Because of same side interior angles theorem, m∠5 +m∠6 = 180°. Substituting gives 110°+ m∠6 =180°, so by the subtraction property of equality , m∠6 = 70°.

5 0
1 year ago
For the binomial expansion of (x + y)^10, the value of k in the term 210x 6y k is a) 6 b) 4 c) 5 d) 7
Sloan [31]

Answer:

a) 6

Step-by-step explanation:

Expanding the polynomial using the formula:

$(x+y)^n=\sum_{k=0}^n \binom{n}{k} x^{n-k} y^k $

Also

$\binom{n}{k}=\frac{n!}{(n-k)!k!}$

I think you mean 210x^6y^4

We can deduce that this term will be located somewhere in the middle. So I will calculate k= 5; k=6 \text{ and } k =7.

For k=5

$\binom{10}{5} (y)^{10-5} (x)^{5}=\frac{10!}{(10-5)! 5!}(y)^{5} (x)^{5}= \frac{10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5! }{5! \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 } \\ =\frac{30240}{120} =252 x^{5} y^{5}$

Note that we actually don't need to do all this process. There's no necessity to calculate the binomial, just x^{n-k} y^k

For k=6

$\binom{10}{6} \left(y\right)^{10-6} \left(x\right)^{6}=\frac{10!}{(10-6)! 6!}\left(y\right)^{4} \left(x\right)^{6}=210 x^{6} y^{4}$

5 0
2 years ago
Your promotional budget for the year is $300 for personal selling, $300 for sales promotion, \$1,500 for advertising, and $500 f
aleksandr82 [10.1K]

Answer:

to get the total promotion budget for the year we will add $300 for personal selling plus $300 for sales promotion plus $1500 for advertising plus $500 for image promotion to the a total of 2600 dollars

8 0
2 years ago
Which is a y-intercept of the continuous function in the table? (0, –6) (–2, 0) (–6, 0) (0, –2)
photoshop1234 [79]

Y- intercept is a point where any graph crosses the y- axis.

X- intercept is a point where any graph crosses the x- axis.

This means the coordinate of the point of intersection will always have the x point as 0. So any point of the form ( 0, y) is the y- intercept. Any point of the form (x,0) is the x- intercept.

Given point are :

(0,-6) : y intercept

(-2,0) : x intercept

(-6,0): x- intercept

(0,-2): y- intercept

3 0
1 year ago
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