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lapo4ka [179]
4 years ago
12

In the figure the horizontal lines are parallel and ab=bc=cd if lm =5 find jm

Mathematics
1 answer:
lisabon 2012 [21]4 years ago
8 0
By the transversal theorem, if segments on a transversal cut by parallel lines are equal, then similar segments cut by the same parallel lines on another transversal are also equal.
Therefore
JK=KL=LM=5
=>
JM=3*5=15
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Find the minimum and maximum of f(x,y,z)=x2+y2+z2 subject to two constraints, x+2y+z=7 and x−y=6.
sweet-ann [11.9K]
Use the method of Lagrange multipliers. We have Lagrangian

L(x,y,z,\lambda_1,\lambda_2)=x^2+y^2+z^2+\lambda_1(x+2y+z-7)+\lambda_2(x-y-6)

with partial derivatives (set equal to 0) of

L_x=2x+\lambda_1+\lambda_2=0
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L_z=2z+\lambda_1=0
L_{\lambda_1}=x+2y+z-7=0
L_{\lambda_2}=x-y-6=0

As x+2y+z=7, and x-y=6, we can obtain

\dfrac12L_x+L_y+\dfrac12L_z=0\implies3\lambda_1-\dfrac12\lambda_2=-7
L_x-L_y=0\implies\lambda_1-2\lambda_2=12
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From this, we find a single critical point:

2x-\dfrac{40}{11}-\dfrac{86}{11}=0\implies x=\dfrac{63}{11}
\dfrac{63}{11}-y=6\implies y=-\dfrac3{11}
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At this point, we have a value of

f\left(\dfrac{63}{11},-\dfrac3{11},\dfrac{20}{11}\right)=\dfrac{398}{11}

To determine what kind of extremum occurs at this point, we check the Hessian of f(x,y,z):

\mathbf H(x,y,z)=\begin{bmatrix}f_{xx}&f_{xy}&f_{xz}\\f_{yx}&f_{yy}&f_{yz}\\f_{zx}&f_{zy}&f_{zz}\end{bmatrix}=\begin{bmatrix}2&0&0\\0&2&0\\0&0&2\end{bmatrix}

We observe that \det\mathbf H(x,y,z)=8>0 at any point (x,y,z), and that the eigenvalues of this matrix are all positive (2 with multiplicity 3), so \mathbf H is positive definite. By the second partial derivative test, this means f(x,y,z) attains a minimum at this critical point. Meanwhile, f has no maximum value.
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3 years ago
What is the volume of the cylinder below?​
daser333 [38]

Answer:

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Step-by-step explanation:

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The radius is 8 and the height is 3

V = pi ( 8)^2 ( 3)

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4 years ago
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write and graph the equation of a line passing through point 1,3 that is parallel and perpendicular to the line with the equatio
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Answer: y=2x+1 is parallel & y=-2x+5 is perpendicular



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