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Helga [31]
2 years ago
11

My room is 7½ ft by 8 ¼ ft. If I want to put a border around it l mid help​

Mathematics
1 answer:
victus00 [196]2 years ago
8 0
9055&43882288273372737373
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Mike reads 40 pages of a book in 50 minutes. David can read 10 more pages than Mike in 50 minutes. How many pages should David b
Sveta_85 [38]
David can read 50 pages in 50 minutes.
(50 pages)/(50 minutes) = 1 page/minute
Since David reads 1 page per minute,
in 80 minutes David can read 80 pages.
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5 0
3 years ago
Which equation represents the radius of a sphere as a function of the volume of the sphere?
boyakko [2]
V ,= 4/3 πr³


solve for r


3V/4=r³

so r is the cubed root of 3V/4
3 0
3 years ago
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
What is the value of x in the equation 4x + 8y = 40, when y = 0.8?
storchak [24]
To find your answer you must substitute y for 0.8
4x + (8)(0.8) = 40
Now solve for x
====================================================================
SOLVING...

<span>4x + (8)(0.8) = 40
</span><span>4x + 6.4 </span>= <span>40
</span>
Subtract 6.4 from each side
<span><span><span>4x</span>+ 6.4 </span>− 6.4 </span>= <span>40 − <span>6.4
</span></span>4x = 33.6

Divide each side by 4
4x ÷ 4 = 33.6 ÷ 4
x = 8.4
3 0
3 years ago
Read 2 more answers
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