Find the directional derivative of f(x, y, z) = xy + yz + zx at p(1, −1, 7) in the direction from p to q(2, 4, 5). duf(1, −1, 7)

Answers

Answer 1

[tex]f(x,y,z)=xy+yz+zx[/tex]

[tex]\implies\nabla f(x,y,z)=(y+z,x+z,y+x)[/tex]

[tex]\vec{pq}=\mathbf q-\mathbf p=(2-1,4-(-1),5-7)=(1,5,-2)[/tex]

[tex]\|\vec{pq}\|=\sqrt{30}[/tex]

[tex]D_{\vec{pq}}f(1,-1,7)=\nabla f(1,-1,7)\cdot\dfrac{(1,5,-2)}{\sqrt{30}}=\dfrac{(6,8,0)\cdot(1,5,-2)}{\sqrt{30}}=\dfrac{46}{\sqrt{30}}[/tex]

Answer 2

Final answer:

The directional derivative of f(x, y, z) = xy + yz + zx at p(1, -1, 7) towards q(2, 4, 5) is found by first calculating the unit vector in the direction from p to q, then the gradient of f at p, and calculating the dot product of the two vectors. The directional derivative comes out to be 16/ √30.

Explanation:

The directional derivative of the function f(x, y, z) = xy + yz + zx at a point p(1, -1, 7) towards the point q(2, 4, 5) can be found by first getting the unit vector in the direction from p to q, then calculating the gradient of the function at p, and finally calculating the dot product of these two vectors (the unit vector and the gradient of the function).

First, calculate the unit vector u = (q - p) / |q - p|. That gives u = (1, 5, -2), and normalize it to get u = 1/ √30 (1, 5, -2).

Next, get the gradient of the function at p(1, -1, 7), which is the vector of first order derivatives. The partial derivatives of f(x, y, z) with respect to x, y, z are y+z, x+z, and x+y respectively. This gives the gradient vector at p as (-8, 8, 0).

Finally, the directional derivative is the dot product of the unit vector and the gradient, which comes out to be 1/ √30 * ((1*-8) + (5*8) + ((-2)*0)) = 16/ √30.

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Related Questions

The number of acres of wetland infested by an invasive plant species after t Years is given by u(t)=12•2^t/3. Solve the equation u(t)=50. What does your answer tell you about the wetland?

Answers

For this case we have the following equation:
u (t) = 12 • 2 ^ t / 3
We equate the equation for u (t) = 50.
We have then:
12 • 2 ^ t / 3 = 50
We clear the value of t:
2 ^ t / 3 = 50/12
log2 (2 ^ t / 3) = log2 (50/12)
t / 3 = log2 (50/12)
t = 3 * log2 (50/12)
t = 6.2 years
Answer:
After 6.2 years 50 acres of wetland are infested by an invasive plant species

Martha is making a scarf for her sister. Each day she knits 1/6 of a scarf....#1 what fraction of the scarf will be complete after 3 day...#2 What fraction will be complete after 6 days....#3 How can you use a number line to prove that your answers are correct?

Answers

You can use the number line to count up by 1/6s like day 1: 1/6 day 2:2/6 and so on. After three days 3/6s of the scarf will be done which is equivalent to 1/2 and on day 6: 6/6 of the scarf will be done which is equivalent to 1.
Hope I could help!
-Montana

PLEASE HELP ME WITH THIS MATH QUESTION

Answers

The general formula of an Arithmetic sequence is:

[tex] a_{n}= a_{1}+(n-1)*d [/tex]

Comparing this formula with the given formula, we can write:

Common Difference = d = 5

Since, every term of an Arithmetic sequence can be obtained by adding the common difference to the previous term, so we can write the recursive formula as:

[tex] a_{n} = a_{n-1} +d[/tex]

For the given sequence we can write:

[tex] x_{n} = x_{n-1} +5[/tex]

So, the answer to this question is 5

square root 2x -5 =11

Answers

Add five to 11 it’ll give you 16 then divide that by 2 it’ll give you 8 than plug it in x=8

solve for x x^2+3x<28

Answers

x² + 3x < 28
x² + 3x - 28 < 0
(x+7)(x-4) < 0

the zeros will be at x = -7 and x =4

∴ the solution of inequality
x ∈ (-7,4)

See the attached figure

Find the volume of the given prism round to the nearest 10th if necessary

Answers

we know that
volume of the prism=area of the base*height
height=24 yd

area of the base
is a equilateral triangle
so
the internal angles are 60 degrees
with the law of sine calculate the area
Area=(1/2)*a*b*sin C-----> (1/2)*13*13*sin 60-----> (1/2)*13*13*√3/2
Area=73.18 yd²

volume of the prism=73.18*24-----> 1756.30 yd³

the answer is
1756.3 yd³

Find the shaded area of the basketball court to the nearest foot.

Answers

So the image consists of 1 rectangle and half a circle.

Rectangle - 12*19 = 228
Half a circle - [tex]\pi r^2 = \pi 6^2 = 36 \pi/2 = 18 \pi [/tex]

In the end there are 2 rectangles (2*228 = 456) and 1 full circle (18*2 = 36), which makes the shaded area 456 ft and 36[tex] \pi [/tex]

First, find the area of the rectangle, which is just 12•19=228 ft.

Then find the area of the semi-circle above it, which is pi times the radius squared. The diameter is 12ft, and we know the radius is half of the diameter, so the radius is 6ft. Our equation is pi times 6^2, or pi times 36.
We’ll just make pi 3.14, so 36•3.14=113.04 feet.

Add the area of the semi-circle and the rectangle together to get one of the shaded areas. 228+113.04= 341.04.

Now that we know the full area of one shaded region, we multiply it by two because there’s two of them. 341.04•2= 682.02.

Since we round it to the nearest foot, our final answer is that the shaded area of the basketball court is 682 square feet.

Solve x^2 +14x = -24 by completing the square. Which is the solution set of the equation?

Answers

Start with x^2 +14x = -24,

Find HALF the value of 14: 7
Square this: 49
Add 49, and then subtract 49: x^2 + 14x + 49 - 49 = -24 We have a
perfect square
Rewrite the quadratic as the sq. of a binomial: trinomial on
the left
(x+7)^2 = 25

Take the sqrt of both sides: x+7 = plus or minus 5

Then x = -2 and x = -12

Answer:

{−12, −2}

Step-by-step explanation:

the surface area of a polyhedron can be found by

Answers

It can be found by summing the areas of its faces

The surface area of a polyhedron is equal to the sum of the area of all of its faces. Said another way, the surface area is the total area covered by the net of a polyhedron.

Answer:

the surface area of a polyhedron can be found by sum of the area of all its sides.

Step-by-step explanation:

The surface area is nothing find the area of each faces and add them together.

Surface area = Sum of area of each faces.

Polyhedron is a three dimensional figure.

For instance, cube a polyhedron which has 6 square faces.

Surface area of a cube = [tex]a^{2} + a^2 + a^2 + a^2 + a^2 + a^2 = 6a^2[/tex], where "a" is the side of the square.

Therefore, the surface area of a polyhedron can be found by sum of the area of all its faces.

hey can you please help me posted picture of question

Answers

Answer:
The solutions are 4 and -4

Explanation:
To get the solutions, we will need to isolate x on one side of the equation.
This can be done as follows:
4x² - 30 = 34
4x² - 30 + 30 = 34 + 30
4x² = 64
x² = 64/4
x² = 16
x = ±√16
either x = +√16 = 4
or x = -√16 = -4

Hope this helps :)

[tex]4x^2-30=34\ \ \ |+30\\\\4x^2=64\ \ \ \ |:4\\\\x^2=16\to x=\pm\sqrt{16}\\\\x=\pm4\\\\\text{Answer:C. x=-4 and D. x=4}[/tex]

Any figure that has line symmetry must also have rotational symmetry. true or false

Answers

A figure that has line symmetry MUST also have a rotational symmetry.

Answer is TRUE.

Andy is 2 years older than Jeff. In 5 years the sum of their ages will be 40.find their ages now? Explain your answer using algebra

Answers

Hello!

You can make two equations based on what you know

let x stand for andy's age
let y stand for Jeff's age

x = 2 + y
5 + x + 5 + y = 40

Put x into the second equation

5 + (2 + y) + 5 + y = 40

Combine like terms

12 + 2y = 40

Subtract 12 from both sides

2y = 28

Divide both sides by 2

y = 14

Put y into the first equation

x = 2 + (14)

x = 16

Andy is 16 and Jeff is 14

Hope this helps!

A rectangular deck is 12 ft by 14 ft. When the length and width are increased by the same amount, the area becomes 288 sq. ft. By how much were the dimensions increased?
2 ft
4 ft
8 ft
10 ft

Answers

To solve this problem you must apply the proccedure shown below:

1. You have the following information given in the problem above:

- The rectangular deck has the following dimensions: 12 ft by 14 ft.
- When the length and width are increased by the same amount, the area becomes 288 ft².

2. Therefore, you have:

A=bxh

288=(12x)(14x)
288=168x²

3. When you clear the x, you obtain:

x=√288/168
x=(2√21)/7≈1.31

Answer:

4 ft option (c)

Step-by-step explanation:

Length of a rectangular deck = 12 ft.

width of rectangular deck= 14 ft.

let the length and width are increased by the x ft.

New length = (12 + x) ft

New width = (14+ x) ft.

area of rectangular width = length × width

according to the question

(12+x)(14+x) = 288

168+[tex]x^{2}[/tex]=+14x+12x = 288

[tex]x^{2}[/tex]+26x = 228-168

[tex]x^{2}[/tex]+26x-120=0

[tex]x^{2}[/tex]+ 30x-4x-120=0

x(x+30)-4x(x+30)

x=4 or x=-30

x=-30 is rejected

Hence, the length and width are increased by the 4 ft.

What is the area of the trapezoid?

56 square inches

240 square inches

108 square inches

120 square inches

Answers

You'd add the bases then multiply by the height to get,
120
Take care ^-^

Answer:

120

Step-by-step explanation:

At a school, 129 students play at least one sport. This is 60% of the students at the school. How many students are at the school?

Answers

(portion of school) = (1.00-portion) x (whole of group)
129 = (1.00-.60) x (whole school)
129 = .4 x (whole school)
129/.4 = Whole school
School size = about 323 students

Rectangle QRST is shown.

What would be a reason to claim that QRST is NOT a square?

SECOND PICTURE ARE THE FOLLOWING SOLUTIONS.

WILL MARK BRAINLIEST!!!

Answers

That would be QT ≠ QR because then QT and QR is not same which makes it impossible for QRST to be square. Square needs all side lengths to be equal.

Remember: P(1+r)t Using the expression above, choose the correct answers for the new balance and the amount of interest earned in the following compound interest problem. $950 at 7% for 8 years, compounded annually. Total Amount = $ Interest Amount = $

Answers

Total amount
950×(1+0.07)^(8)
=1,632.28

Interest amount
1,632.28−950
=682.28

Suppose angle A is complementary to angle B, and angle B is supplementary to angle C. If measure of angle A is 21 degree, find measure of angle C.

Answers

C = 111 degrees

Since A and B are complementary they add to 90. So we know that B must be 69 degrees.

Since B and C are supplementary they add to 180. So we know that C must be 111.

Final answer:

In solving this problem, we first identify that angle B is 69 degrees as it is complementary to 21-degree angle A (adding to 90 degrees). Then, we find angle C to be 111 degrees as it is supplementary to angle B (adding to 180 degrees).

Explanation:

In this problem, we are told that angle A (measuring 21 degrees) is complementary to angle B. Complementary angles are angles that add up to 90 degrees. So, we can find the measurement of angle B by subtracting the measurement of angle A from 90 degrees, which gives us 90 - 21 = 69 degrees for angle B.

Next, we are told angle B is supplementary to angle C. Supplementary angles are angles that add up to 180 degrees. So, we can find the measurement of angle C by subtracting the measurement of angle B from 180 degrees, which results in 180 - 69 = 111 degrees for angle C.

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hey can you please help me posted picture of question

Answers

G(x) can be obtained from F(x) by shifting F(x) 4 units to the right and 1 unit upwards.

Shift towards right can be obtained by subtracting 4 from the x in the function and the upward shift can be obtained by adding 1 to the function.

So,

G(x) = F(x-4) + 1

Thus,

G(x) = (x-4)² + 1

So, the correct answer to this question is option B

Find the slope and y-intercept fotv

Answers

Look at the picture.

[tex]A(3;\ 2)\\\\the\ slope:\ m=\dfrac{2}{3}\\\\B(0; 0):\ y-intercept[/tex]

A rectangular picture has dimensions 2 1/2 by 1 1/2. It is to be enlarged so that it is still similar to the original rectangle. If the longer dimension is now 10 , how much longer than the original is the perimeter now

Answers

The original perimeter is:
p = 2 (w) +2 (l)
First we look for the scale factor of the longest dimension, which is given by:
k = (10) / (2 1/2)
k = 4
Then, the perimeter is given by:
p '= 2 (4w) +2 (4l)
Rewriting:
p '= 4 (2 (w) +2 (l))
p '= 4p
Answer:
the perimeter now is 4 times longer than the original

hey can you please help me posted picture of question

Answers

The parent reciprocal function is represented in the table.

As you can see, increasing the value of x, decreases the value of y as both are inversely related as:

[tex]y= \frac{1}{x} [/tex]

We can also write this equation as:

yx = 1

This means, the product of each x and y value must be 1 for parent reciprocal function. If you observe the table, the product of each pair is 1.

So the correct answer is option A

The pairs of values of x and y above represent that of the equation,

y = 1/x.

If we are to substitute the given values of x to determine the values of y for checking, the results are as shown below.

1. y = 1/-2 = -0.5
2. y = 1/-1 = -1
3. y = 1/2 = 0.5
4. y = 1/3 = 0.33

Thus, the answer to this item is the first option, y = 1/x.

In a clinical trial for a new headache medication, participants are randomly assigned to a treatment group or a placebo group. They do not know whether they are receiving the medication or a placebo. The element of experiment design being used is _____.

randomization

blinding

replication

Answers

This is called randomization

Answer:

The element of experiment design being used is blinding.

Step-by-step explanation:

In a clinical trial for a new headache medication, participants are randomly assigned to a treatment group or a placebo group.

The first step is to define the treatment group, it is performed in a random way, as most of the statically experiments.

They do not know whether they are receiving the medication or a placebo.

People inside both groups do not have information about the medication. It means they are blind to the possible effects.

The blinding method is used mostly to remove the placebo effect in the sample. Because of that it is commonly used for clinical or any experimental assays as it was showed in this example.

(4+8r^5) - (4r^5 + 2)

Answers

(4+8r^5) - (4r^5 + 2)

= 4+8r^5 - 4r^5 - 2
= 4r^5 +2
= 2(2r^5 + 1)

The simplest form of the equation is 2(2r^5 + 1)

When two births are randomly selected, the sample space for genders is bb, bg, gb and gg. assume that those four outcomes are equally likely. does the mean of the sample proportions equal the population proportion of girls in two births?

Answers

The answer is No. The mean of the sample and that of the population can not be equal. As the sample moves closer to the population, the mean of sample moves closer to the mean of the population

PLEASE PLEASE PLEASE HELP ME WITH ALGEBRA 1 PLEASE!?!?!?!?!!!?!?!?!?!?! I WILL GIVE LOTS OF POINTS AND BRAINLEST PLEEEEEEEEEEEEEEEEASE!!?!?!?!?!?!!?!?!?!?

What are the benefits and limitations to each method? When is it best to use each method?
1) graphing 2) substitution method 3) elimination method

Answers

With graphs on paper you can visually see intersecting lines(one solution), identical lines
(infinite solutions), or parallel lines(no solutions).
This method works best for lines that have integers for solutions. The task of determining rational or irrational solutions from graphs is difficult.

The substitution method usually must be used for equations with higher exponents. The deawback of this method is having to work with fractions.

The elimination method is often the method of choice.
This method however requires that you learn to recognize when the systems produce parallel lines or have equations that represent the same lines.

Answer:

don't copy exactly, this is my answer to the question.

Step-by-step explanation:

what is the next number in this pattern 0.54,0.6,0.66,0.72

Answers

This is a linear pattern. In this case, every next number goes up by 0.06.
So, add 0.06 to 0.72.
0.06+0.72=0.78.
So, your answer is 0.78.
Have a nice day! ♪

Ivanova has a healthcare plan with $25 co-pays to see doctors, $10 co-pays for prescription drugs, and an annual deductible of $1,500. In February she has a broken ankle which costs $2,300 in hospital treatment. As a follow-up in March she sees three doctors who bill $657, $134, and $232 and buys two prescriptions which cost $12 and $27. What is Ivanova's share of the costs for March?

Answers

1842 I think ,hope this helps :D

Ok, so let's start with the hospital bill. The total is $2,300, but she has a $1,500 deductible. So subtract $1,500 from $2,300. You should get $800. Now the doctor's visits. She has a $25 co-pay, and she saw three doctors. Multiply $25 by 3. You should have $75. Lastly, the prescriptions. She has a $10 co-pay, and there are 2 prescriptions, so multiply 2 and $10. You get $20. Now add the hospital bill, the doctor's bills, and the prescriptions.

$800+$75+$20=$895.

Jamal states that ax + b = a(x + c), given a, b, and c are not equal to 0.

What must be the value of c for Jamal’s statement to be true?

Answers

To answer the question, we must find c of the equation:
ax + b = a (x + c)
If we divide by "a" on both sides we have to:
x + b / a = x + c
For values of "a" other than zero.
So:
c = b / a.
So that the affirmation of Jamal Be true c = b / a

Answer:

[tex]c=\frac{b}{a}[/tex]

Step-by-step explanation:

We have been given an equation [tex]ax+b=a(x+c)[/tex], given that a, b and c are not equal to 0.

Let us solve for the value of c that will make Jamal's statement true.

Upon dividing both sides of our equation by a we will get,

[tex]\frac{ax+b}{a}=\frac{a(x+c)}{a}[/tex]

[tex]\frac{ax+b}{a}=x+c[/tex]

[tex]\frac{ax}{a}+\frac{b}{a}=x+c[/tex]

[tex]x+\frac{b}{a}=x+c[/tex]

Now, we will subtract x from both sides of our equation.

[tex]x-x+\frac{b}{a}=x-x+c[/tex]

[tex]\frac{b}{a}=c[/tex]

Therefore, the value of [tex]c=\frac{b}{a}[/tex] will make Jamal’s statement true.

Approximately how many boxes that are 2 ft. by 2 ft. by 3.5 ft. fit inside of the back of a tractor trailer box truck whose dimensions are 28 ft. by 8 ft. by 9 ft.?

26 Boxes

87 Boxes

144 Boxes

216 Boxes

Answers

144 is the correct answer

Approximately how many boxes that are 2 ft. by 2 ft. by 3.5 ft. fit inside of the back of a tractor trailer box truck whose dimensions are 28 ft. by 8 ft. by 9 ft.?

Solution:

Volume of trailer box= Number of boxes * Volume of Box

Volume of Trailer Box=Length*Width*Height

Dimensions of Trailer Box are 28 ft. by 8 ft. by 9 ft.

Volume of Trailer Box=28*8*9ft³=2016ft³

Dimensions of box are 2 ft. by 2 ft. by 3.5 ft.

Volume of Box=2*2*3.5ft³=14ft³

Volume of Trailer Box=Number of Boxes *Volume of Box

2016=Number of Boxes*14

To get, Number of Boxes, Let us divide by 14

2016/14=Number of Boxes=144

Number of Boxes=144

Answer: Option C:144 boxes

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4.5 divided by 0.5+0.1 in computation words "Am I going to find my treasure?" he asked.He stuck his hand into the pouch, and felt around for one of the stones. As he did so, both of them pushed through a hole in the pouch and fell to the ground. The boy had never even noticed that there was a hole in his pouch. He knelt down to find Urim and Thummim and put them back in the pouch. But as he saw them lying there on the ground, another phrase came to his mind."Learn to recognize omens, and follow them," the old king had said.How does the symbol of the stones affect the meaning in this passage?By symbolizing the boys journey and the choices he will make, the stones remind the boy to pay attention to the omens.The stones symbolize the omens and the trials the boy will face, making him fearful of the future.The stones symbolize the boys treasure, reminding him of the great wealth that he will earn.By symbolizing the magical forces of the king of Salem, the stones represent the sorcery the boy will use to find his treasure. A very challenging job for new presidents is to ______. question 25 options:a.prepare their first executive budgetb.move into the white housec.prepare and deliver their first state of the union addressd.nominate and gain confirmation for their cabinet and hundreds of other officials what happens when sulphur trioxide reacts with concentrated sulphur acid ? One important factor with respect to children coping with adversity is a child's Which sentence accurately uses the homophone stationary? The handwriting and the stationary were absolutely exquisite. Her mother told her to remain stationary while she was inside. I think I will buy some new stationary before I write to her. Perhaps the invitations should be printed on stationary. Which of the following is not a valid probability?A 0.3B 1C 0B 106 The capital asset pricing model is used to calculate the effect of increase in prices of capital assets due to inflation. a. True b. False At the end of "Marriage is a Private Affair", which change in the story's atmosphere, marked by lightening and thunder, relates to the father's change of heart toward his son's marriage? What was Maryland Eastern Shore during slavery? Calculate the freezing point of a solution that contains 8.0 g of sucrose (c12h22o11) in 100 g of h2o. kf for h2o = 1.86c/m Penelope finally believes that the man who has killed the suitors really is her husband only after discussion of what object? Slave state legislators had more power in what part of Congress during the 1850s? A. The House of Representatives B. The Supreme Court C. The Senate D. The Executive list 3 types of active transport and 3 types of passive tranport. (5a-5)(2a+2)Find each product AT a groyrey store salted peanuts in a shell cost 30 cents per ounce is 5$ enough money to buy 1 pound of peanuts if it is what amount of money will be leaft over Why was 1928 considered a bull market year? BRAINLIEST please help?A graph that displays bivariate data is called _________ ?Positive AssociationStrength of AssociationBivariate DataScatterplotTwo-Way TableTwo-Way Relative Frequency TableNo AssociationRelative FrequencyAssociationNegative Association A man has 28 coins in his pocket, all of which are dimes and quarters. if the total value of his change is 520 cents, how many dimes and how many quarters does he have? a hiker first hike down into a canyon 345 ft below sea level she then hike on her husband 50 ft of the side of the mountain what was the final altitude of the hiker