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3 years ago

Please help!! 15 points

Mathematics

1 answer:

zlopas [31]3 years ago

3 0

Answer:

? = 125°

Step-by-step explanation:

The central angle is equal to the measure of the arc that subtends it, thus

? = 125°

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If f(x, y, z) = x sin(yz), (a) find the gradient of f and (b) find the directional derivative of f at (2, 4, 0) in the direction

valentina_108 [34]

Answer:

a) $\nabla f(x,y,z) = \sin{yz}\mathbf{i} + xz\cos{yz}\mathbf{j} + xy \cos{yz}\mathbf{k}$ .

b) $Du_{f}(2,4,0) = -\frac{8}{\sqrt{11}}$

Step-by-step explanation:

Given a function $f(x,y,z)$ , this function has the following gradient:

$\nabla f(x,y,z) = f_{x}(x,y,z)\mathbf{i} + f_{y}(x,y,z)\mathbf{j} + f_{z}(x,y,z)\mathbf{k}$ .

(a) find the gradient of f

We have that $f(x,y,z) = x\sin{yz}$ . So

$f_{x}(x,y,z) = \sin{yz}$

$f_{y}(x,y,z) = xz\cos{yz}$

$f_{z}(x,y,z) = xy \cos{yz}$ .

$\nabla f(x,y,z) = f_{x}(x,y,z)\mathbf{i} + f_{y}(x,y,z)\mathbf{j} + f_{z}(x,y,z)\mathbf{k}$ .

$\nabla f(x,y,z) = \sin{yz}\mathbf{i} + xz\cos{yz}\mathbf{j} + xy \cos{yz}\mathbf{k}$

(b) find the directional derivative of f at (2, 4, 0) in the direction of v = i + 3j − k.

The directional derivate is the scalar product between the gradient at (2,4,0) and the unit vector of v.

We have that:

$\nabla f(x,y,z) = \sin{yz}\mathbf{i} + xz\cos{yz}\mathbf{j} + xy \cos{yz}\mathbf{k}$

$\nabla f(2,4,0) = \sin{0}\mathbf{i} + 0\cos{0}\mathbf{j} + 8 \cos{0}\mathbf{k}$ .

$\nabla f(2,4,0) = 0i+0j+8k=(0,0,8)$

The vector is $v = i + 3j - k = (1,3,-1)$

To use v as an unitary vector, we divide each component of v by the norm of v.

$|v| = \sqrt{1^{2} + 3^{2} + (-1)^{2}} = \sqrt{11}$

$v_{u} = (\frac{1}{\sqrt{11}}, \frac{3}{\sqrt{11}}, \frac{-1}{\sqrt{11}})$

Now, we can calculate the scalar product that is the directional derivative.

$Du_{f}(2,4,0) = (0,0,8).(\frac{1}{\sqrt{11}}, \frac{3}{\sqrt{11}}, \frac{-1}{\sqrt{11}}) = -\frac{8}{\sqrt{11}}$

6 0

3 years ago

What is 9 ounces equal to in pounds? please answer in fractions:)

Tema [17]

Answer:

9/16

Step-by-step explanation:

First, note that 10000 multiplied by .5625 will get rid of the decimal point. Therefore, make a fraction where the numerator is .5625 times 10000 and denominator is 10000 like this:

.5625 x 10000

10000

Next, multiply the numbers in the numerator together and keep the denominator as is. Now our fraction looks like this:

5625 /10000

The greatest common factor of 5625 and 10000 is 625, which means you can divide the numerator and denominator by 625 and keep the same value:

5625 ÷ 625 /10000 ÷ 625

And when calculating the numerator and denominator in our fraction above, we get .5625 as a fraction in the simplest form possible:

.5625 =

9 /16

6 0

3 years ago

Read 2 more answers

Learn with an example

liubo4ka [24]

Answer:12

Step-by-step explanation:

36:3=12

6 0

2 years ago

Read 2 more answers

Calculate the slope of the line going through A(-4,3) and B(0,6)

Whitepunk [10]

Answer:

3/4

Step-by-step explanation:

The slope of the line is given by

m= (y2-y1)/(x2-x1)

= (6-3)/(0 - -4)

= (6-3)/( 0+4)

3/4

5 0

3 years ago

Find the total travel time: Depart 5:30 pm MST Arrive 9:20 pm CST

Len [333]

Answer:

2 hours 50 minutes

Step-by-step explanation:

Mountain Standard Time is 1 hour earlier than Central Standard Time, so the departure time is equivalent to 6:30 pm in CST.

The time between 6:30 pm and 9:20 pm is 2 hours 50 minutes.

8 0

3 years ago