How full of water is the glass? 2/3 1/3 2/1 1/2

Find equations of the tangent plane and the normal line to the given surface at the specified point. x + y + z = 8exyz, (0, 0, 8

Dima020 [189]

Let $f(x,y,z)=x+y+z-8e^{xyz}$ . The tangent plane to the surface at (0, 0, 8) is

$\nabla f(0,0,8)\cdot(x,y,z-8)=0$

The gradient is

$\nabla f(x,y,z)=\left(1-8yze^{xyz},1-8xze^{xyz},1-8xye^{xyz}\right)$

so the tangent plane's equation is

$(1,1,1)\cdot(x,y,z-8)=0\implies x+y+(z-8)=0\implies x+y+z=8$

The normal vector to the plane at (0, 0, 8) is the same as the gradient of the surface at this point, (1, 1, 1). We can get all points along the line containing this vector by scaling the vector by $t$ , then ensure it passes through (0, 0, 8) by translating the line so that it does. Then the line has parametric equation

$(1,1,1)t+(0,0,8)=(t,t,t+8)$

or $x(t)=t$ , $y(t)=t$ , and $z(t)=t+8$ .

(See the attached plot; the given surface is orange, (0, 0, 8) is the black point, the tangent plane is blue, and the red line is the normal at this point)

4 0

3 years ago

What is the equation, in point slope form, of the line

melamori03 [73]

Answer:

y=3/2x+11/2

Step-by-step explanation:

Hello! Sorry I just saw this

Anyways, let's continue

first, we need to find the equation of the line with the one that is (-2,-4) and (2,2)

first, we need to find the slope

the equation for slope is y2-y1/x2-x1

so let's label the points

x1=-2

y1=-4

x2=2

y2=2

now plug it in

2-(-4)/2-(-2)=6/4=3/2

now, let's turn it into a line

the point-slope form is y-y1=m(x-x1) (m=slope)

now, plug it in

y-(-4)=3/2(x-(-2))

simplify to

y+4=3/2(x+2)

turn into y=mx+b format

y+4=3/2x+3

subtract 4 on both sides

y=3/2x-1

Now for the line that is parallel.

Parallel lines have the same slopes, so you automatically know that the new line will be y=3/2x+b

To make sure (-3,1) is a solution to the point, put 1 as y and -3 as x

1=3/2(-3)+b

1=-9/2+b

add 9/2 on both sides

b=11/2 or 5.5

now, put it into the equation

y=3/2x+11/2

Hope this helps!

5 0

3 years ago

6 miles in 24 minutes. find the average rate in miles per hour

melamori03 [73]

Answer:

15 Miles Per Hour

Step-by-step explanation:

The speed is 0.25 miles per minute, so if the person goes at a steady speed, its 15 MPH

7 0

3 years ago

Read 2 more answers

How do you do series with an n

SOVA2 [1]

The problem can be solved step by step, if we know certain basic rules of summation. Following rules assume summation limits are identical.
$\sum{a+b}=\sum{a}+\sum{b}$
$\sum{kx}=k\sum{x}$
$\sum_{r=1}^n{1}=n$
$\sum_{r=1}^n{r}=n(n+1)/2$

Armed with the above rules, we can split up the summation into simple terms:
$\sum_{r=1}^n{40r-21n+8}=n$
$=40\sum_{r=1}^n{r}-21n\sum_{r=1}^n{1}+8\sum_{r=1}^n{1}$
$=40\frac{n(n+1)}{2}-21n^2+8n$
$=20n(n+1)-21n^2+8n$
$=28n-n^2$

=> (a) f(x)=28n-n^2
=> f'(x)=28-2n
=> at f'(x)=0 => x=14
Since f''(x)=-2 <0 therefore f(14) is a maximum
(b) f(x) is a maximum when n=14

(c) the maximum value of f(x) is f(14)=196

4 0

2 years ago

Help help d. e. f. g. h.

andrey2020 [161]

D. 7
e. 4
f. 5
g. 11

Get these by adding/subtracting the number at the end (use the opposite operation) and then divide the coefficient from both sides.

7 0

3 years ago