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Nata [24]
3 years ago
7

A box of multi-color paper clips contains 25 red clips, 15 blue clips, 15 green clips and 20 pink clips. If Roy

Mathematics
1 answer:
kolbaska11 [484]3 years ago
3 0

Answer:

8/15

Step-by-step explanation:

Probability is the ratio of the number of possible outcome to the number of total outcome. The probability that an event will happen added to the probability that the event will not happen gives 1. In other words, the outcome of a probability cannot exceed 1.

The probability that an event a will happen or that another independent event  b will happen is the sum of the probability that a will happen to the probability that b will happen.

Given that there are 25 red clips, 15 blue clips, 15 green clips and 20 pink clips,

total number of clips available

= 25 + 15 + 15 + 20

= 75

probability of picking a

red paper clip = 25/75

blue paper clip = 15/75

Probability of selecting red or  blue paper clip

= 25/75 + 15/75

= 40/75

= 8/15

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The plane x+y+2z=8 intersects the paraboloid z=x2+y2 in an ellipse. Find the points on this ellipse that are nearest to and fart
DiKsa [7]

Answer:

The minimum distance of   √((195-19√33)/8)  occurs at  ((-1+√33)/4; (-1+√33)/4; (17-√33)/4)  and the maximum distance of  √((195+19√33)/8)  occurs at (-(1+√33)/4; - (1+√33)/4; (17+√33)/4)

Step-by-step explanation:

Here, the two constraints are

g (x, y, z) = x + y + 2z − 8  

and  

h (x, y, z) = x ² + y² − z.

Any critical  point that we find during the Lagrange multiplier process will satisfy both of these constraints, so we  actually don’t need to find an explicit equation for the ellipse that is their intersection.

Suppose that (x, y, z) is any point that satisfies both of the constraints (and hence is on the ellipse.)

Then the distance from (x, y, z) to the origin is given by

√((x − 0)² + (y − 0)² + (z − 0)² ).

This expression (and its partial derivatives) would be cumbersome to work with, so we will find the the extrema  of the square of the distance. Thus, our objective function is

f(x, y, z) = x ² + y ² + z ²

and

∇f = (2x, 2y, 2z )

λ∇g = (λ, λ, 2λ)

µ∇h = (2µx, 2µy, −µ)

Thus the system we need to solve for (x, y, z) is

                           2x = λ + 2µx                         (1)

                           2y = λ + 2µy                       (2)

                           2z = 2λ − µ                          (3)

                           x + y + 2z = 8                      (4)

                           x ² + y ² − z = 0                     (5)

Subtracting (2) from (1) and factoring gives

                     2 (x − y) = 2µ (x − y)

so µ = 1  whenever x ≠ y. Substituting µ = 1 into (1) gives us λ = 0 and substituting µ = 1 and λ = 0  into (3) gives us  2z = −1  and thus z = − 1 /2 . Subtituting z = − 1 /2  into (4) and (5) gives us

                            x + y − 9 = 0

                         x ² + y ² +  1 /2  = 0

however, x ² + y ² +  1 /2  = 0  has no solution. Thus we must have x = y.

Since we now know x = y, (4) and (5) become

2x + 2z = 8

2x  ² − z = 0

so

z = 4 − x

z = 2x²

Combining these together gives us  2x²  = 4 − x , so

2x²  + x − 4 = 0 which has solutions

x =  (-1+√33)/4

and

x = -(1+√33)/4.

Further substitution yeilds the critical points  

((-1+√33)/4; (-1+√33)/4; (17-√33)/4)   and

(-(1+√33)/4; - (1+√33)/4; (17+√33)/4).

Substituting these into our  objective function gives us

f((-1+√33)/4; (-1+√33)/4; (17-√33)/4) = (195-19√33)/8

f(-(1+√33)/4; - (1+√33)/4; (17+√33)/4) = (195+19√33)/8

Thus minimum distance of   √((195-19√33)/8)  occurs at  ((-1+√33)/4; (-1+√33)/4; (17-√33)/4)  and the maximum distance of  √((195+19√33)/8)  occurs at (-(1+√33)/4; - (1+√33)/4; (17+√33)/4)

4 0
3 years ago
..........................
Rainbow [258]
Option A

x+8=-3
x+8-8=-3-8
x=-11


Hope I didn't mess up for your sake!
7 0
3 years ago
If R= {(1,2), (3,4), (5,6), (8,9)} find the domain and range of R a. Find value of X and Y.​
podryga [215]
<h2>The Domain and Range of a Set of Ordered Pairs</h2><h3>Answer:</h3>

Domain: x \in \{ 1, 3, 5, 8 \}

Range: y \in \{ 2, 4, 6, 9 \}

<h3>Step-by-step explanation:</h3>

We are given the set of ordered pairs: R. The Domain is just the set containing all the x values of each ordered pair of R. Recall that in the ordered pair (a,b), a is the x value. In set R, we can see that the x values are \{ 1, 3, 5, 8 \}.

For the Range, it is just the set containing all the y values of each ordered pair of R. Recall that in the ordered pair (a,b), b is the y value. In set R, we can see that the y values are \{ 2, 4, 6, 9 \}.

7 0
2 years ago
The following is the frequency distribution for the speed of a sample of automobiles traveling on an interstate highway. Speed (
svlad2 [7]

Answer:

Standard Deviation is 18.57 .

Step-by-step explanation:

We are given the frequency distribution for the speed of a sample of automobiles traveling on an interstate highway;

  Speed (mph)     Frequency (f)      X         X*f           X - Xbar        (X-Xbar)^{2}

     50 - 54                    4                52       208         52 - 65 = -13         169

     55 - 59                    3                57        171           57 - 65 = -8           64

     60 - 64                    2                62        124          62 - 65 = -3            9

     65 - 69                    5                67        335          67 - 65 = 2             4

     70 - 74                     2                72        144           72 - 65 = 7            49

     75 - 79               <u>      5       </u>         77     <u>   385    </u>      77 - 65 = 12          144

                                <u> ∑f = 21   </u>              ∑X*<u>f = 1367 </u>            

Mean of the data, Xbar = \frac{\sum Xf}{\sum f}

                                       = \frac{1367}{21} = 65.09 ≈ 65 .        

Now, Standard deviation, s = \sqrt{\frac{\sum f*(X-Xbar)^{2} }{n-1}}

s =  \sqrt{\frac{(4*169)+(3*64)+(2*9)+(5*4)+(2*49)+(5*144)}{6-1} } = \sqrt{\frac{1724}{5} } = 18.57

Therefore, standard deviation is 18.57 .

           

5 0
2 years ago
Use log to solve for x: 38^2x=109
Ksju [112]
38^2x = 109
Apply lg on both sides
lg 38^2x = lg 109
Bring the power down
2x lg 38 = lg 109
2x = lg 109/lg 38
x = (lg 109/lg 38) ÷ 2
= 0.645

Hope it helped!
8 0
3 years ago
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