Find an equation of the plane that passes through the point (1, 3, 4) and cuts off the smallest volume in the first octant.

12x +4y + 3z=36

Step-by-step explanation:

The equation of plane is given by

z-zo = a(x-xo) + b(y-yo)

pass through (1,3,4)

Z -4 = a(x -1) +b(y-3)

The question is asking us to optimize a and b. To minimize the volume V both a and b should be negative as the normal vector should be towards the negative x and y direction so that a finite tetrahedron can be formed in the first octant.

we need x , y and z intercepts o define volume

x intercept( y, z =0) =

y intercept (x, z =0) =

z intercept ( x, y =0) = -(a+3b-4)

Base =

Volume =

Volume(a, b) =

now we differentiate partially in terms to a and b the volume to minimize and get a and b.

ΔV(a, b) = ,  = 0

Taking the first part of differential it will give

b(a+3b-4) [3a -(a+3b -4)] =0

(a+3b-4) because the volume will become zero if this becomes true

2a -3b = -4  ..................(1)

similarly the second part of the differential will give

a-6b=4 ................(2)

on solving 1 and 2 we get

a = -4 and b = -4/3

so the equation will be

Z -4 = -4(x -1) - 4/3*(y-3)

final equation

12x +4y + 3z=36

answer: 48

just remember to use pemdas:

parentheses

expressions

multiplication

division

addition

subtraction

find the magnitude of the trig term  

csc

(

x

)

−

1

by taking the absolute value of the coefficient.

1

find the lower bound of the range.

tap for more

y

=

−

2

find the upper bound of the range.

tap for more

y

=

0

the range is  

y

≤

−

2

or  

y

≥

0

.

(

−

∞

,

−

2

]

∪

[

0

,

∞

)

{

y

|

y

≤

−

2

,

y

≥

0

}

step-by-step explanation: