Mathematics: The radius of a circle is 4 kilometers. Use 3.14 for . What is the circle s circumference?

The point on the plane determines the volume of the box, since . Restricting the box to lie within the first octant is to say that .Let s do it via Lagrange multipliers. The Lagrangian iswith partial derivatives (set equal to 0)We haveWe already assume , so we can ignore those options, leaving us with , , and . Substituting into the plane equation, we getSo the box with largest volume has its vertex (the one opposite the vertex at the origin) in the plane at , giving a volume of ....

The radius of a circle is 4 kilometers. Use 3.14

Ответ:

alexamorantess

31.08.2022

Mathematics

The point

on the plane x+2y+3z=9

determines the volume of the box, since V(x,y,z)=xyz

. Restricting the box to lie within the first octant is to say that x,y,z0

.

Let's do it via Lagrange multipliers. The Lagrangian is

$L(x,y,z,\lambda)=xyz+\lambda(x+2y+3z-9)$

with partial derivatives (set equal to 0)

$L_x=yz+\lambda=0$
$L_y=xz+2\lambda=0$
$L_z=xy+3\lambda=0$
$L_\lambda=x+2y+3z-9=0$

We have

$L_y-2L_x=xz-2yz=z(x-2y)=0\implies z=0\text{ or }x=2y$
$L_z-3L_x=xy-3yz=y(x-3z)=0\implies y=0\text{ or }x=3z$
$2L_z-3L_y=2xy-3xz=x(2y-3z)=0\implies x=0\text{ or }2y=3z$

We already assume x,y,z0

, so we can ignore those options, leaving us with x=x

, $y=\dfrac x2$ , and $z=\dfrac x3$ . Substituting into the plane equation, we get

$x+2\dfrac x2+3\dfrac x3=3x=9\implies x=3\implies y=\dfrac32\text{ and }z=1$

So the box with largest volume has its vertex (the one opposite the vertex at the origin) in the plane at $\left(3,\dfrac32,1\right)$ , giving a volume of $\dfrac92$ .

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The radius of a circle is 4 kilometers. Use 3.14 for ​. What is the circle's circumference?

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The radius of a circle is 4 kilometers. Use 3.14 for . What is the circle's circumference?