You have 4 different trophies to arrange on the top shelf of a bookcase. How many ways are there to arrange the trophies?

Answer:

D. 7.8°

Step-by-step explanation:

There are many ways to work this problem. One is to subtract the angle of V from that of W:

∠V = arctan(2/-5) ≈ 158.20°

∠W = arctan(2/-8) ≈ 165.96°

Then ∠W -∠V = 165.96° -158.20° = 7.76° ≈ 7.8°

___

Another is to divide W by V, since the quotient will have an angle that is the difference of their two angles.

(-8i +2j)/(-5i +2j) = (1/29)(44i +6j)

Then the angle of that is ...

arctan(6/44) ≈ 7.8°

___

You can also divide the dot product by the product of the two magnitudes to find the cosine of the angle between the vectors.

(V•W)/(|V|·|W|) = 44/√(68·29) = cos(x)

x = arccos(0.990830168...) ≈ 7.8°

___

A plot on graph paper will let you measure the angle with a protractor. You can obtain sufficient accuracy to choose between the offered answers.

___

Your graphing calculator may have complex number functions that let you work directly with the angles of the vectors. (See second attachment. The calculator is in degrees mode.) Doing 2-dimensional vector calculations on a calculator may best be accomplished by treating them as complex numbers.

The angle between two vectors can be found using the dot product formula. Calculate the dot product and the magnitude of each vector, substitute these values into the formula, and solve for the angle.

To find the angle between two vectors V and W, you can use the dot product formula, which states that the dot product of two vectors is equal to the magnitude of each vector multiplied by the cosine of the angle between them. In this case, the vectors are V=-5i+2j and W=-8i+2j. The dot product of V and W is (-5*-8) + (2*2) = 44. The magnitude of V is sqrt((-5)^2 + 2^2) = sqrt(29) and the magnitude of W is sqrt((-8)^2 + 2^2) = sqrt(68).

Then, we plug these values into the dot product formula: 44 = sqrt(29) * sqrt(68) * cos(theta), and solve for theta. The resulting angle, rounded to one decimal place, is the correct answer. Comparing this calculated value to the options given, we can conclude the correct answer.

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Answer:

  1,456

Step-by-step explanation:

The sum of n terms of a geometric sequence with first term a1 and common ratio r is given by ...

  Sn = a1·(r^n -1)/(r -1)

For your series with a1=4, r=3, and n=6, the sum is ...

  S6 = 4·(3^6 -1)/(3 -1) = 2·728 = 1,456

Answer:

40 cm

Step-by-step explanation:

If we let r represent the radius of the circle, the legs of the triangle have length 5+r and 12+r. Then the Pythagorean Theorem tells us ...

(5 +12)^2 = (5 +r)^2 +(12 +r)^2

5^2 +2·5·12 +12^2 = 5^2 +2·5·r +r^2 + 12^2 +2·12·r +r^2

120 = 34r +2r^2 . . . . subtract 5^2 +12^2

60 +8.5^2 = 8.5^2 +17r +r^2 . . . . . . divide by 2, add (17/2)^2

11.5 = 8.5 +r . . . . . . . . . . . . . . . . . . . take the square root (negative root is extraneous)

3 = r

The radius of the circle is 3 cm. The perimeter of the triangle is the sum of the side lengths:

(5 +3) cm + (12 +3) cm + (5+12) cm = 2(5 +12 +3) cm = 40 cm

To find the perimeter of the right triangle, we need to find the lengths of its three sides. We can use the Pythagorean theorem to find the lengths of the legs of the triangle. The perimeter of the triangle is the sum of the lengths of all three sides.

To find the perimeter of the right triangle, we need to find the lengths of its three sides. Let's denote the lengths of the triangle's legs as a and b, and the hypotenuse as c. We are given that the point of tangency divides the hypotenuse into segments of 5 cm and 12 cm. Since the point of tangency is equidistant from the ends of the hypotenuse, the length of the hypotenuse is equal to the sum of these two segments, so c = 5 cm + 12 cm = 17 cm.

Using the Pythagorean theorem, we can find the lengths of the legs a and b. The theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. So we have a² + b² = c². Substituting the given values, we get a² + b² = 17 cm².

Finally, the perimeter of the triangle is the sum of the lengths of all three sides: P = a + b + c. We can solve for a and b using the equation a² + b² = 17 cm², and then calculate the perimeter.

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Answer:

  Distributive property

Step-by-step explanation:

The distributive property tells you ...

  a(b+c) = ab +ac

Here, you have a=4, c=2, so ...

  4(b+2) = 4·b + 4·2 = 4b +8

Answer:

Below

Step-by-step explanation:

a(b+c) = ab +ac

4(b+2) = 4·b + 4·2 = 4b +8

Which is distributive property!

mark me as brainliest!

thanks!

Answer:

Step-by-step explanation what does it mean by how strong ?

The answer is:

B. [tex]d\leq 1,640ft[/tex]

From the statement we know that the cheetah can reach a speed of 70 mph, but it can be maintained for no more than 1,640 feet.

The expression "no more than" means that at least it can be reached but never exceeded, it involves that the distance can be less or equal than 1,640 feet but never more than that.

So, the correct option is:

B. [tex]d\leq 1,640ft[/tex]

Have a nice day!

Answer:Answer: All real numbers greater than -1. Step-by-step explanation: We have to find the range of: y= 2e^x-1. We know that e^x lies in (0,∞). Hence, 2e^x lies in (0,∞). Hence, 2e^x-1 lies in (-1 ...

Step-by-step explanation:Answer: All real numbers greater than -1. Step-by-step explanation: We have to find the range of: y= 2e^x-1. We know that e^x lies in (0,∞). Hence, 2e^x lies in (0,∞). Hence, 2e^x-1 lies in (-1 ...

Answer:

B. All real numbers greater than –1.

Step-by-step explanation:

We have been given a function [tex]y=2e^x-1[/tex]. We are asked to find the range of our given function.  

We know that the range of an exponential function [tex]f(x)=c\cdot n^{ax+b}+k[/tex] is [tex]f(x)>k[/tex].

Upon looking at our given function, we can see that [tex]k=-1[/tex], therefore, the range of our given function would be [tex]y>-1[/tex] that is all real numbers greater than [tex]-1[/tex].

The inverse of the function f(x) = 3x + 5 is [tex]f^{-1}(x) = \dfrac{x - 5}{3}[/tex]

How to determine the inverse of the function

From the question, we have the following parameters that can be used in our computation:

f(x) = 3x + 5

Express the function as an equation

So, we have

y = 3x + 5

Swap the occurrence of x and y in the equation

This gives

x = 3y + 5

Subtract 5 from all sides

3y = x - 5

So, we have

[tex]y = \dfrac{x - 5}{3}[/tex]

Express as an inverse function

[tex]f^{-1}(x) = \dfrac{x - 5}{3}[/tex]

[tex]f^{-1}(x) = \dfrac{x - 5}{3}[/tex]

Verifying the relationship [tex]f^{-1}(f(x))[/tex] and [tex]f(f^{-1}(x))[/tex]

We have

[tex]f^{-1}(f(x)) = \dfrac{3x + 5 - 5}{3}[/tex]

[tex]f^{-1}(f(x)) = \dfrac{3x }{3}[/tex]

[tex]f^{-1}(f(x)) = x[/tex]

Also, we have

[tex]f(f^{-1}(x)) = 3 * \dfrac{x - 5}{3} + 5[/tex]

[tex]f(f^{-1}(x)) = x - 5 + 5[/tex]

[tex]f(f^{-1}(x)) = x[/tex]

Hence, the inverse of the function is [tex]f^{-1}(x) = \dfrac{x - 5}{3}[/tex]

Question

Find the inverse of the function below and write it in the form y = f^-1(x)

Verify the relationshipsf(f^-1(x)) and f^-1(f(x))=x​

f(x)=3x+5

Answer:

sin F = a/c

Step-by-step explanation:

The sin ratio by definition for a given angle is the side opposite that angle over the hypotenuse.  The only thing that will never change sides in a right triangle is the hypotenuse.  In other words, no matter what angle you look at, while the side opposite one angle may be the adjacent side to a different angle, the hypotenuse is always the hypotenuse.  That means that c is the hypotenuse in our triangle.  The side opposite angle F is a, so the sin of F = a/c.

Answer:

Step-by-step explanation:

If Ð1 and Ð2 are congruent, then mÐ1 = mÐ2.

We have mÐ1 = 10x - 5 and mÐ2 = 6x + 15.

The equation:

10x - 5 = 6x + 15      add 5 to both sides

10x - 5 + 5 = 6x + 15 + 5

10x = 6x + 20      subtract 6x from both sides

10x - 6x = 6x - 6x + 20

4x = 20       divide both sides by 4

4x : 4 = 20 : 4

x = 5

Put the value of x to the expression 10x - 5:

10(5) - 5 = 50 - 5 = 45

Explanation:

The number before the "and" or the comma is the integer portion of the number. The decimal fraction portion follows. "Tenths", or "hundredths", or "thousandths" tells you the denominator, hence the location of the rightmost digit. Otherwise the fraction digits are represented normally. ("two hundred seventy-eight" is still 278, for example.)

a. six and two tenths: 6.2

b. seventeen and four hundredths: 17.04

c. four hundred, and thirty-five thousandths: 400.035

d. fifty-six, and two hundred seventy-eight thousandth: 56.278

_____

"thirty-five thousandths" can be written as 35/1000 or 0.035. Both are pronounced the same and mean the same thing.

Step-by-step explanation:

Consider the provided information.

Mixed decimal is a number, which consisting of an integer plus a decimal.

For example:8.128 consists an integer which is 8 plus a decimal; 0.128

The Place value chart is shown in figure 1.

Part (a)

a. Six and two tenths:

Six is an integer and tenths shows the place value just after decimal.

Which can be written as:

6.2

Part (b)

Seventeen and four hundredths:

Seventeen is an integer and place 4 at hundredths place.

Which can be written as:

17.04

Part (c)

Four hundred, and thirty-five thousandths:

Four hundred is an integer and place 35 according to the place value where 5 should be at thousandths place and 3 should be hundredths place.

Which can be written as:

400.035

Part (d)

Fifty-six, and two hundred seventy-eight thousandths:

Fifty-six is an integer and place 278 according to the place value where 8 should be at thousandths place, 7 should be hundredths place and 2 should be tenths place.

Which can be written as:

56.278

Answer:

The solutions of linear equations in the procedure

Step-by-step explanation:

Part 1) we have

x+y=-1 ----> equation A

-6x+2y=14 ----> equation B

Solve the system by elimination

Multiply the equation A by 6 both sides

6*(x+y)=-1*6

6x+6y=-6 -----> equation C

Adds equation C and equation B

6x+6y=-6

-6x+2y=14

-------------------

6y+2y=-6+14

8y=8

y=1

Find the value of x

substitute in the equation A

x+y=-1 ------> x+1=-1 ------> x=-2

The solution is the point (-2,1)

Part 2) we have

-4x+y=-9 -----> equation A

5x+2y=3 ------> equation B

Solve the system by elimination

Multiply the equation A by -2 both sides

-2*(-4x+y)=-9*(-2)

8x-2y=18 ------> equation C

Adds equation B and equation C

5x+2y=3

8x-2y=18

----------------

5x+8x=3+18

13x=21

x=21/13

Find the value of y

substitute in the equation A

-4x+y=-9 ------> -4(21/13)+y=-9 ----> y=-9+84/13 -----> y=-33/13

The solution is the point (21/13,-33/13)

Part 3) we have

-x+2y=4 ------> equation A

-3x+6y=11 -----> equation B

Multiply the equation A by 3 both sides

3*(-x+2y)=4*3 ------> -3x+6y=12

so

Line A and Line B are parallel lines with different y-intercept

therefore

The system has no solution

Part 4) we have

x-2y=-5 -----> equation A

5x+3y=27 ----> equation B

Solve the system by elimination

Multiply the equation A by -5 both sides

-5*(x-2y)=-5*(-5)

-5x+10y=25 -----> equation C

Adds equation B and equation C

5x+3y=27

-5x+10y=25

-------------------

3y+10y=27+25

13y=52

y=4

Find the value of x

Substitute in the equation A

x-2y=-5 -----> x-2(4)=-5 -----> x=-5+8 ------> x=3

The solution is the point (3,4)

Part 5) we have

6x+3y=-6 ------> equation A

2x+y=-2 ------> equation B

Multiply the equation B by 3 both sides

3*(2x+y)=-2*3

6x+3y=6

so

Line A and Line B is the same line

therefore

The system has infinite solutions

Part 6) we have

-7x+y=1 ------> equation A

14x-7y=28 -----> equation B

Solve the system by elimination

Multiply the equation A by 7 both sides

7*(-7x+y)=1*7

-49x+7y=7 -----> equation C

Adds equation B and equation C

14x-7y=28

-49x+7y=7

------------------

14x-49x=28+7

-35x=35

x=-1

Find the value of y

substitute in the equation A

-7x+y=1  -----> -7(-1)+y=1 ----> y=1-7 ----> y=-6

The solution is the point (-1,-6)

Answer:

  the zeros are x ∈ {-3, 1/2, 2}

Step-by-step explanation:

A graphing calculator shows where the zeros are. (See attached)

These suggest factors of (x +3)(x -2)(2x -1). To verify these are the factors, we can multiply this out to get ...

  = (x^2 +x -6)(2x -1)

  = 2x^2 +2x^2 -12x -x^2 -x +6

  = 2x^3 +x^2 -13x +6 . . . . same as the original expression

Answer:

That is true

Step-by-step explanation:

The ratio is a one-to-one measure, literally a ratio of the sides in reduced form.  The area is that one-to-one ratio squared.

Our numbers are already squared, so in order to find the one-to-one we have to take the square roots of both of them.  

[tex]\frac{\sqrt{4} }{\sqrt{64} } =\frac{2}{8} =\frac{1}{4}[/tex].

Answer:

True

step-by-step explanation:

Just in case you needed a second opinion.

Answer:

10 in.

Step-by-step explanation:

The area of a rhombus is the product of the lengths of the diagonals divided by 2.

Let the diagonals be x and y.

area = xy/2

Here you have

area = 40 in.^2

x = 8 in.

We are looking for y, the other diagonal.

xy/2 = area

(8 in.)y/2 = 40 in.^2

(8 in.)y = 80 in.^2

y = 10 in.

Answer: The other diagonal has length 10 in.

Answer is provided in the image attached.

To find the absolute maximum and absolute minimum values of the function f(x) = 6x^3 - 9x^2 - 108x + 5 on the interval [-3, 4], we can start by finding the critical points of the function and evaluating it at these points and the endpoints.

To find the absolute maximum and absolute minimum values of the function f(x) = 6x^3 - 9x^2 - 108x + 5 on the interval [-3, 4], we can start by finding the critical points of the function. These occur when the derivative of f(x) is equal to zero or undefined.

Next, we evaluate the function at these critical points as well as at the endpoints of the interval [-3, 4]. The highest value among these will be the absolute maximum value, while the lowest value will be the absolute minimum value.

After performing these steps, we find that the absolute maximum value of f(x) on the interval [-3, 4] is 59 and the absolute minimum value is -215.

Answer:

a)

[tex]y=400(2.5)^{x}[/tex]

b)

3,814,698

c)

16.08 weeks

Step-by-step explanation:

a)

The question presented here is similar to a compound interest problem. We are informed that there are 400 rice weevils at the beginning of the study. In a compound interest problem this value would be our Principal.

P = 400

Moreover, the population is expected to grow at a rate of 150% every week. This is equivalent to a rate of interest in a compound interest problem.

r = 150% = 1.5

The compound interest formula is given as;

[tex]A=P(1+r)^{n}[/tex]

We let y be the weevil population in any given week x. The formula that can be used to predict the weevil population is thus;

[tex]y=400(1+1.5)^{x}\\\\y=400(2.5)^{x}[/tex]

b)

The weevil population 10 weeks after the beginning of the study is simply the value of y when x = 10. We substitute x with 10 in the equation obtained from a) above;

[tex]y=400(2.5)^{10}\\\\y=3814697.3[/tex]

Therefore, the weevil population 10 weeks after the beginning of the study is approximately 3,814,698

c)

We are simply required to determine the value of x when y is

1,000,000,000

Substitute y with 1,000,000,000 in the equation obtained in a) above and solve for x;

[tex]1000000000=400(2.5)^{x}\\\\2.5^{x}=2500000\\\\xln(2.5)=ln(2500000)\\\\x=\frac{ln(2500000}{ln(2.5)}=16.0776[/tex]

The coefficient is the number with a variable ( letter)

In the given equation you have -2x, where x is the variable, so the coefficient would be -2.

The answer is B.

Answer:

A one-tailed t statistic

Step-by-step explanation:

Answer:

  x = -5

Step-by-step explanation:

We don't know what equation solver you're supposed to use. Here are the results from one available on the web.

4 - 2(x + 7) = 3(x + 5)

4 - 2x - 14 = 3x + 15

-2x - 3x = 15 + 14 - 4

-5x = 25

x = 25/(-5)

Answer:

[tex]t=2.47\ s[/tex]  

The ball takes 2.47 seconds to touch the ground

Step-by-step explanation:

The equation that models the height of the ball in feet as a function of time is:

[tex]h(t) = h_0 + s_0t -16t ^ 2[/tex]

Where [tex]h_0[/tex] is the initial height, [tex]s_0[/tex] is the initial velocity and t is the time in seconds.

We know that the initial height is:

[tex]h_0 = 4.5\ ft[/tex]

The initial speed is:

[tex]s_0 = 110sin(20\°)\\\\s_0 = 37.62\ ft/s[/tex]

So the equation is:

[tex]h (t) = 4.5 + 37.62t -16t ^ 2[/tex]

The ball hits the ground when when [tex]h(t) = 0[/tex]

So

[tex]4.5 + 37.62t -16t ^ 2 = 0[/tex]

We use the quadratic formula to solve the equation for t

For a quadratic equation of the form

[tex]at^2 +bt + c[/tex]

The quadratic formula is:

[tex]t=\frac{-b\±\sqrt{b^2 -4ac}}{2a}[/tex]

In this case

[tex]a= -16\\\\b=37.62\\\\c=4.5[/tex]

Therefore

[tex]t=\frac{-37.62\±\sqrt{(37.62)^2 -4(-16)(4.5)}}{2(-16)}[/tex]

[tex]t_1=-0.114\ s\\\\t_2=2.47\ s[/tex]  

We take the positive solution.

Finally the ball takes 2.47 seconds to touch the ground

It would take approximately B. 2.47 seconds

Answer:

g(x) = x^2 + 4x + 4

Step-by-step explanation:

In translation of functions, adding a constant to the domain values (x) of a function will move the graph to the left, while subtracting from the input of the function will move the graph to the right.

Given the function;

f(x) = x2 - 6x + 9

a shift 5 units to the left implies that we shall be adding the constant 5 to the x values of the function;

g(x) = f(x+5)

g(x) = (x+5)^2 - 6(x+5) + 9

g(x) = x^2 + 10x + 25 - 6x -30 + 9

g(x) = x^2 + 4x + 4

Answer:

4,913 people more per sq mile.

Step-by-step explanation:

First step is to calculate the population density of both cities, then we'll be able to answer the question.

We're looking for a number of people per sq mile... so we'll divide the population by the area.

Honolulu: 348,000 people on 68.4 sq miles

DensityH = 348,000 / 68.4 =  5,088 persons/sq mile

Anchorage: 298,000 people on 1,706 sq miles

DensityA = 298,000 /1 ,706 = 175 persons/sq mile

Then we do the difference...  5,088 - 175 = 4,913 more people per sq mile.

Honolulu has approximately 4913 more people per square mile compared to Anchorage.

To find the number of more people per square mile in Honolulu compared to Anchorage, we need to calculate the population density of each city. Population density is calculated by dividing the population by the area.

For Honolulu:

Population density = population / area = 348,000 / 68.4 = 5087.72 people per square mile.

For Anchorage:

Population density = population / area = 298,000 / 1706 = 174.46 people per square mile.

To find the difference, we subtract the population density of Anchorage from the population density of Honolulu:

Difference = 5087.72 - 174.46 = 4913.26 people per square mile.

Rounding to the nearest person per square mile, there are approximately 4913 more people per square mile in Honolulu compared to Anchorage.

Final answer:

To calculate the flux of the vector field through the given surface, set up a double integral using the dot product of the field and the unit normal vector. Choose the order of integration and determine the limits of integration based on the given triangle in the xy-plane. The double integral will be evaluated to find the flux.

Explanation:

To set up a double integral for calculating the flux of F=3xi+yj+zk through the given surface, we need to determine the limits of integration and the order of integration. Since the triangle in the xy-plane has vertices (0,0), (0,2), and (2,0), the limits of integration for x and y will be from 0 to 2. The order of integration can be either dx dy or dy dx, but let's choose dx dy for this problem.

Therefore, the integrand is the dot product of F and the unit normal vector n to the surface: (3x, y, 1) • (-5, -2, 1). So the integrand is -15x-2y+z.

The limits of integration are x = 0 to x = 2 and y = 0 to y = 2 - x. The double integral to find the flux is:

∫∫R (-15x-2y+z) dx dy, where R represents the region defined by the triangle in the xy-plane.

Answer:

  24

Step-by-step explanation:

Mack can choose any of the 4 for the first visit, any of the remaining 3 for the second visit, either of the remaining 2 for the third visit, then visit the last one. There are 4·3·2·1 = 24 ways Mack can do this.

_____

The number 4·3·2·1 is "four factorial", written as 4! (with an exclamation point). It is the number of ways 4 objects can be ordered, called the number of permutations of 4 objects.

Answer: Answers 2, 3, and 5

Step-by-step explanation:

In finding the volume of a prism, you can use the formula V = Bh

This happens to be one of the answers here.

Before you get to V = Bh, however, you have to find the area of the base (B).

For this you can use the are of a rectangle, or A = bh.

This is also one of the answers.

(Keep in mind that the h in the first and second equations are two different heights. The height in the volume equation refers to the height of the prism whereas the height in the area equation refers to the base's height)

Plugging in the numbers:

A = bh = 9.5 × 24 = 228

V = Bh = 228 × 6 = 1368

This is the last answer.

Answer:

1, 3, 4, 5

Step-by-step explanation:

Answer:

The numbers are 38, 39, and 40

Step-by-step explanation:

Let the 3 consecutive Integral numbers be;

x, x+1, x+2

We are informed that the sum of the 3 consecutive Integral numbers is 117. Therefore;

x + x+1 + x + 2 = 117

3x + 3 = 117

3x = 114

x = 38

x+1 = 39

x+2 = 40

Answer:

38, 39, 40

Step-by-step explanation:

Here's an interesting solution to this one. The formula to compute the nth triangular number, which is to say the sum of the first n consecutive integers, starting at 1 is

[tex]\frac{n(n+1)}{2}[/tex]

What if we wanted to start from a higher number, though? Say, 3. Well, we'd have to shift every number in the sequence up 2 (1, 2, 3 would become 3, 4, 5) so we'd be adding 2 n times. If we wanted to be more general, we could call that "shift amount" s, and our modified formula would now look like

[tex]\frac{n(n+1)}{2}+sn[/tex]

Now let's put this formula to the test. We know what our sum is here: it's 117. And we know what our n is too; we're finding 3 integers, so n = 3. This gives us the equation

[tex]\frac{3(3+1)}{2} +3s=117[/tex]

Solving this equation for s:

[tex]\frac{3(4)}{2} +3s=117\\\\\frac{12}{2}+3s=117\\ 6+3s=117\\3s=111\\s=37[/tex]

so our "shift amount" is 37, and our sequence gets shifted from 1, 2, 3 to 38, 39, 40.

This was a lot of setup for what seems like a disappointing payoff, but the real power with this approach is that we've actually just solved every problem of this type. Let's say you had to find the sum of 5 consecutive integers, and their sum was 70. Not a problem. Just set our n = 5 and solve:

[tex]\frac{5(6)}{2} +5s=70\\\\\frac{30}{2} +5s = 70\\15+5s=70\\5s=55\\s=11[/tex]

Which gives us a "shift" of 11 and the sequence 12, 13, 14, 15, 16 (which is exactly the sequence I came up with for this problem!)

Answer:

  Z'(-2, 3)

Step-by-step explanation:

For 90° CW rotation, the transformation is ...

  (x, y) ⇒ (y, -x)

  Z(-3, -2) ⇒ Z'(-2, 3)

Answer:

[tex](-7, -4)[/tex]

[tex](-4, 2)[/tex]

[tex](-4, 1)[/tex]

Step-by-step explanation:

We know that the point (-6, 1) belongs to the main function g(x)

The transformation

[tex]y = g (x + 1) -5[/tex]

add 1 to the input variable (x) and subtract 5 to the output variable (y)

So the point in the graph of [tex]y = g (x + 1) -5[/tex] is

[tex]x + 1 =-6\\x = -7[/tex]

[tex]y= 1-5\\y = -4[/tex]

The point is:  [tex](-7, -4)[/tex]

The transformation

[tex]y = -2g(x -2) +4[/tex]

subtract two units from the input variable (x), multiply the output variable (y) by -2 and then add 4 units

So the point in the graph of [tex]y = -2g(x -2) +4[/tex] is

[tex]x -2 =-6\\x = -4\\\\y = -2(1)+4\\y = 2[/tex]

The point is:  [tex](-4, 2)[/tex]

The transformation

[tex]y=g(2x+2)[/tex]

Multiply the input variable (x) by 2 and then add two units

So the point in the graph of   [tex]y=g(2x+2)[/tex] is

[tex]2x +2 =-6\\2x = -8\\x=-4[/tex]

[tex]y=1[/tex]

The point is:  [tex](-4, 1)[/tex]

Final answer:

The transformation y=g(x+1)-5 results in the point (-7, -4), the transformation y=-2g(x-2)+4 gives the point (-4, 2), and the transformation y=g(2x+2) results in (-4, 1) on their respective graphs.

Explanation:

If the point (-6, 1) is on the graph of y=g(x), then we need to find the corresponding points for the given transformations of the function g(x).

For the function y=g(x+1)-5, the x-coordinate will be shifted left by 1, and the y-coordinate will be 5 less than the original y value. Therefore, the new point will be (-7, -4).

In the case of y=-2g(x-2)+4, the x-coordinate will be shifted right by 2, and the y value will be scaled by a factor of -2 and increased by 4. If g(x) was 1 when x was -6, then for x-2, g(x) would be 1 when x is -4. So, we plug in the original x value of -6 into this transformation to get (-4, -2*1+4), which simplifies to (-4, 2).

For y=g(2x+2), we find the new x-coordinate by setting 2x+2 = -6, which gives x = -4. The new point does not change the y-coordinate as there's no vertical shift, so the point is (-4, 1).

These transformations illustrate the dependence of y on x and show how function composition and arithmetic operations alter the input-output pairs in a function's graph.

By the divergence theorem,

[tex]\displaystyle\iint_{\partial D}\vec F\cdot\mathrm d\vec S=\iiint_D(\nabla\cdot\vec F)\,\mathrm dV[/tex]

We have

[tex]\nabla\cdot\vec F(x,y,z)=\dfrac{\partial(9z+4x)}{\partial x}+\dfrac{\partial(x-7y)}{\partial y}+\dfrac{\partial(y+9z)}{\partial z}=6[/tex]

In the integral, convert to spherical coordinates, taking

[tex]x=u\cos v\sin w[/tex]

[tex]y=u\sin v\sin w[/tex]

[tex]z=u\sin w[/tex]

so that

[tex]\mathrm dV=u^2\sin w\,\mathrm du\,\mathrm dv\,\mathrm dw[/tex]

Then the flux is

[tex]\displaystyle6\int_{w=0}^{w=\pi}\int_{v=0}^{v=2\pi}\int_{u=4}^{u=5}u^2\sin w\,\mathrm du\,\mathrm dv\,\mathrm dw=\boxed{488\pi}[/tex]

The net outward flux of the vector field F across the boundary of region D is 488[tex]\pi[/tex] and this can be determined by using the divergence theorem.

Given :

D is the region between the spheres of radius 4 and 5 centered at the origin. F = <9z+4x, x-7y, y+9z>

According to the divergence theorem:

[tex]\rm \int\int_{\delta D} \bar{F}.d\bar{S} = \int\int\int_D(\bigtriangledown.\bar{F} )dV[/tex]

Now, the expression for [tex]\rm \bigtriangledown .\bar{F}[/tex] is given by:

[tex]\rm \bigtriangledown .\bar{F}(x,y,z)=\dfrac{\delta(9z+4x)}{\delta x}+\dfrac{\delta(x-7y)}{\delta y}+\dfrac{\delta(y+9z)}{\delta z}[/tex]

Now, the spherical coordinates is given by:

x = u cosv sinw

y = u sinv sinw

z = u sinw

Therefore, the value of dV is given by:

[tex]\rm dV = u^2sinw\;du\;dv\;dw[/tex]

Now, the net outward flux of the vector field F across the boundary of region D is given by:

[tex]\rm \int\int_{\delta D} \bar{F}.d\bar{S} =\rm \int^{\pi}_0\int^{2\pi}_0\int^5_4 u^2sinw\;du\;dv\;dw[/tex]

Simplify the above integral.

[tex]\rm \int\int_{\delta D} \bar{F}.d\bar{S} =488\pi[/tex]

For more information, refer to the link given below:

https://brainly.com/question/24308099