Given f(x)= raiz 6x and g(x)= 1/x-6 Which value is in the domain of f g

Answers

Answer 1

The domain of  the composition function fog(x) when f(x)= √6x and g(x)= 1/ (x-6) is equal to  (-∞, 6) U (6, ∞).

Function f(x) = √6x

And g(x)= 1/ (x-6

Composition function fog is ,

fog(x) = f(g(x))

Domain of fog(x),

First, find the domain of g(x),

Here,

g(x) = 1/(x-6)

The denominator of g(x) cannot be equal to zero,

So we must exclude x = 6 from the domain of g(x).

The domain of g(x) is equal to ,

domain of g(x) = {x | x ≠ 6}

The range of g(x) is,

As x approaches 6 from the left, g(x) approaches negative infinity,

And as x approaches 6 from the right, g(x) approaches positive infinity.

Range of g(x) = (-∞, 0) U (0, ∞)

Now, the domain of fog(x),

fog(x)

= f(g(x))

= √6g(x)

=√ 6 (1 /(x - 6 ) )

Square root sign must be non-negative.

Since the range of g(x) does not include zero,

Exclude the part of the range that is less than zero.

The domain of fog(x) is,

domain of fog(x)

= {x | g(x) > 0}

= {x | x < 6} U {x | x > 6}

The domain of fog(x) consists of all real numbers except x = 6.

Therefore, the domain of fog(x) is (-∞, 6) U (6, ∞).

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The above question is incomplete, the complete question is :

Given f(x)= √6x and g(x)= 1/ (x-6) Which value is in the domain of fog.


Related Questions

The endpoints of AB are given. Find the coordinates of the midpoint M. Then find AB. A(-3, 5), B(3, 5) ​

Answers

Answer:

Step-by-step explanation:

The difference between the x coordinates is 6, (difference between -3 and 3). Half of 6 is 3. Add this to the lowest value (-3) to get to zero.

Repeating the exercise for the Y coordinates we get a difference of 0 (there's no difference between the values 5 & 5). So adding this difference (0) to the lowest value (5) we get 5 still.

So the midpoint is (0,5)

Two firms producing identical products engage in price competition. Cost of firm 1 is $20 per unit produced and cost of firm 2 is $15 per unit produced. There are no fixed costs. Firms produce only after they learn the quantity demanded. Each firm can choose any real number in the interval [15,25] as its price.
For tie-breaking we will assume that if both firms set the same price, all consumers purchase from firm 2.
The payoff/profit function of firm 1 is:
(p1 - 20)(100 - p1) if p1 is less than or equal to p2,
0 if p1 is greater than p2
The payoff/profit function of firm 2 is:
(p2 - 15)(100 - p2) if p2 is less than p1,
0 if p2 is greater than or equal to p1
Given all of this information, solve the following parts of the problem:
a) is p1 = 20, p2 = 19.50 a Nash Equilibrium?
b) is there a Nash Equilibrium in which Firm 2 makes a positive profit?
c) How many strategies does player 1 have?
d) is p1 = 15, p2 = 15 a Nash Equilibrium?
e) is p1 = 21, p2 = 21 a Nash Equilibrium?

Answers

a) No, p1 = 20, p2 = 19.50 is not a Nash Equilibrium.

b) Yes, there is a Nash Equilibrium in which Firm 2 makes a positive profit.

c) Player 1 has infinitely many strategies.

d) Yes, p1 = 15, p2 = 15 is a Nash Equilibrium.

e) No, p1 = 21, p2 = 21 is not a Nash Equilibrium.

What is Nash Equilibrium?

Nash Equilibrium is a state of a strategic game where no player has an incentive to deviate from his or her chosen strategy after considering the strategies of other players. A game has more than one Nash equilibrium if players are unable to agree on a cooperative strategy to play.

Finding Nash Equilibrium

a) Is p1 = 20, p2 = 19.50 a Nash Equilibrium?No. Firm 1 has an incentive to decrease the price to 19.49, thus breaking the tie in its favour. So p1=20, p2=19.5 is not a Nash equilibrium.b) Is there a Nash Equilibrium in which Firm 2 makes a positive profit?Yes. There are several equilibria in which Firm 2 makes a positive profit. One such equilibrium is when both firms charge the same price of 15, at which both firms earn a profit of 375.c) How many strategies does player 1 have?Player 1 has infinitely many strategies to choose from as they can choose any real number in the interval [15,25] as their price.d) Is p1 = 15, p2 = 15 a Nash Equilibrium?Yes. This is a Nash Equilibrium because neither firm has an incentive to change their strategy as they are earning non-zero profits. Both firms earn a profit of 375.e) Is p1 = 21, p2 = 21 Nash Equilibrium?No. If Firm 1 changes its price to 20.99, its profit increases from 405 to 407.99. Therefore, p1=21 and p2=21 is not a Nash Equilibrium.

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Triangle HEF is the image of triangle FGH after a 180 degree rotation around point K. Select all statements that must be true.

Answers

Answer: can you share the picture?

Step-by-step explanation:

.

Download
Stations - Applicatio....pdf
2.) If you add Ginger and her
grandmother's
ages together it
would be 66. Her grandmouer
is ten times older than Ginger.
What are the ages of each
person?

Answers

Answer:

Step-by-step explanation:

Let us consider the Ginger to be x years old.

Therefore, her grandmother will be 10x years old(10 times Ginger's age)

So,

10x+x=66

(adding both Ginger and her grandmother's age)

11x=66

(adding x and 10x)

x=66/11

x=6

Ginger's age is 6.

Her grandmother is 10 times her age.

Therefore she is 60 years old.

Which pair of functions are inverses of each other?

Which pair of functions are inverses of each other?

Answers

Answer:

c

Step-by-step explanation:

the inverse of something cubed is the cubed root of that number

“To find 623, I can just take the value of 613 then either multiply it by 12 or divide it by 2.” Do you agree with her? Explain why or why not.

Answers

Step-by-step explanation:

613 and 623 are very close values.

multiplying by 12 or a division by 2 changes the starting value drastically.

so, this cannot be right.

and dividing 613 by 2 also makes it even smaller. but to get 623 we need to make it bigger.

so, therefore, even without actually calculating the numbers, the statement is proven false.

Which system of equations has no solutions?

Which system of equations has no solutions?

Answers

Answer:

C

Step-by-step explanation:

Causes of variation that can be identified and eliminated are called what?

Answers

The causes of variation that can be identified and eliminated are called; Assignable Causes.

What are the causes of Variation?

There are two primary causes of variation in the quality of a product or process. These two primary causes are called;

Common causes.Assignable causes.

Now, Common causes of variation are defined as random causes that we cannot identify. However, Assignable causes of variation are those that can be identified and eliminated.

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I NEED HELP!!!

The mean number of passengers on a flight from Los Angeles to San Francisco is 82. The plane holds a maximum of 102 passengers. List the possible number of passengers on the flight over the past 5 days.

Answers

The possible number of passengers on the flight over the past 5 days can range from 72 to 92 passengers

Since the mean number of passengers on a flight from Los Angeles to San Francisco is 82, we can assume that this is the average number of passengers over a period of time. To list the possible number of passengers on the flight over the past 5 days, we can use the range of values that are within one standard deviation of the mean.

Assuming a normal distribution of passengers on the flight, we can use the empirical rule to estimate the range of values within one standard deviation of the mean. The empirical rule states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean.

Therefore, the range of possible values for the number of passengers on the flight can be estimated as:

Mean - Standard deviation <= Number of passengers <= Mean + Standard deviation

Since we do not have the standard deviation of the data, we can use a conservative estimate of the standard deviation as 10% of the maximum capacity of the plane, which is 102 passengers. Therefore, the standard deviation can be estimated as 10.2 passengers.

Substituting these values into the formula, we get:

82 - 10.2 <= Number of passengers <= 82 + 10.2

Simplifying this inequality, we get:

71.8 <= Number of passengers <= 92.2

Therefore, the possible number of passengers on the flight over the past 5 days can range from 72 to 92 passengers, assuming a normal distribution of passengers and a standard deviation of 10.2 passengers.

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During the first basketball game of the year, you made 12 baskets. In the second game of the year, you only made 9. What is the percent decrease in the shots made from the first game to the second?

Answers

Answer:

25%

Step-by-step explanation:

75% of 12 is 9, meaning you got 75% of the shots in.

which means you missed 25% of the shots

25%

explanation:

9 is 75% of 12, meaning u got 75% of the shots. if you went from 100 to 75% of the shots, the decrease would be 25% because that is the difference between 75 and 100.

What is the value of s^2t for s = 7 and t = 20? as a whole number

Answers

Answer:

7^40 is umm I don't know sorry

Step-by-step explanation:

But hopefully it's helpful.

Answer:

steps below

Step-by-step explanation:

if your problem is s²ⁿ = s²*²⁰ = s⁴⁰     (n replace your t)

s²ⁿ = 7⁴⁰ = 6366805760909027985741435139224001

if the problem is s² x 20 = 7² x 20 = 980

Any help please? Having a hard time

Any help please? Having a hard time

Answers

Answer:

the answer is c

Step-by-step explanation:

x and y have to be at least 12, so that's why the greater than sign was used

the amount of money had to be at most $230, so that's why the less than sign was used

they said x and y represent black and white respectively, so that means x is black and y is white, so it is 18x and 20y

Answer:

x+y ≥ 12

18x+20y ≤ 230

Step-by-step explanation:

At least means that is the smallest number they can buy, so they must buy at least 12

x+y ≥ 12

no more than 230 means that is the maximum amount they can spend

18x+20y ≤ 230

Please help me I got quarantine and do not know what we are doing

Please help me I got quarantine and do not know what we are doing

Answers

the slope is how many units its goes up and how many it goes right basically rise/run

The confidence associated with an interval estimate is called the:.

Answers

The confidence associated with an interval estimate is called the confidence level.

When making an interval estimate, such as estimating a population parameter based on a sample, it is important to provide an indication of the level of confidence we have in the estimate. The confidence level represents the probability or level of certainty that the true population parameter falls within the calculated interval.

For example, if we calculate a 95% confidence interval for the mean height of a population, it means that we are 95% confident that the true population mean height lies within that interval. This implies that if we were to repeat the sampling and interval estimation process multiple times, approximately 95% of the intervals would contain the true population parameter.

The confidence level is typically expressed as a percentage, such as 90%, 95%, or 99%. A higher confidence level indicates a greater level of certainty in the interval estimate but also results in a wider interval.

In summary, the confidence level associated with an interval estimate represents the level of confidence or probability that the interval contains the true population parameter.

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1 EUR = 317 huf and 1 EUR= 1.0 GBP. How much is HUF/GBP exchange
rates ?

Answers

The HUF/GBP exchange rate is 0.00315 or 1 GBP = 317.46 HUF (rounded to two decimal places). To find the HUF/GBP exchange rates, we need to first convert HUF to GBP using the given exchange rates.1 EUR = 317 HUF and 1 EUR = 1.0 GBP.

We can use these exchange rates to find the HUF/GBP exchange rate as follows:

HUF/GBP = (HUF/EUR) / (GBP/EUR)

To find HUF/EUR, we can use the first exchange rate:

HUF/EUR = 1 / 317

To find GBP/EUR, we can take the reciprocal of the second exchange rate:

GBP/EUR = 1 / 1.0

Therefore, HUF/GBP = (HUF/EUR) / (GBP/EUR)

= (1/317) / (1/1.0)

= 0.00315 GBP/HUF

So, the HUF/GBP exchange rate is 0.00315 or 1 GBP = 317.46 HUF (rounded to two decimal places).

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(please help!!!!) The list represents a student's grades on tests in their math class.

59, 65, 70, 80, 98, 71, 45, 79, 77, 85

Find the range for the data set.

45
53
74
98

Answers

Answer:

Range is 53

Step-by-step explanation:

The range is the difference between the maximum and minimum values in a data set.

The minimum value in the data set is 45, and the maximum value is 98.

Therefore, the range is:

98 - 45 = 53

So the answer is 53.

2. Find the slope, or rate of change, represented in the table below. x -1 0 1 2 3 У -8 3 14 25 36​

Answers

Answer:

11

Step-by-step explanation:

slope is change in y over change in x. so, if x is increasing by one every time and y is increasing by 11, then the slope is 11/1, which we write as 11.

The area of a rectangle is 192 in ^2. The ratio of the length to the width is 4:3. Find the length and the width.

The length of the rectangle is ___ in.
The width of the rectangle is___.

Answers

Answer:

Length = 16 in

Width = 12 in

Step-by-step explanation:

Given:

Area of a rectangle = 192 in²

Ratio(length to width) = 4:3

Find:

Length and the width

Computation:

Ratio(length to width) = 4:3

So,

Length = 4a

Width = 3a

Area of a rectangle = Length x Width

192 = 4a x 3a

192 = 12a²

a² = 16

a = 4

Length = 4a = 4 x 4 = 16 in

Width = 3a = 3 x 4 = 12 in

If A = 5 and B = 3, what will be displayed when code corresponding to the following pseudocode is run? (In the answer options, new lines are separated by commas.)
Do
Write A^2
Set A = A - 1
While A >= B

Answers

The output when the given pseudocode is executed with A = 5 and B = 3 will be "25, 16, 9, 4, 1".

The given pseudocode includes a loop that iterates as long as A is greater than or equal to B. In each iteration, the square of A is displayed, and A is decremented by 1. We are asked to determine the output when A is initially 5 and B is 3.

Step 1: Initialization

A is set to 5 and B is set to 3.

Step 2: Iteration 1

Since A (5) is greater than or equal to B (3), the loop executes.

The square of A (5²) is displayed, resulting in the output "25".

A is decremented by 1, so A becomes 4.

Step 3: Iteration 2

A (4) is still greater than or equal to B (3).

The square of A (4²) is displayed, resulting in the output "16".

A is decremented by 1, so A becomes 3.

Step 4: Iteration 3

A (3) is still greater than or equal to B (3).

The square of A (3²) is displayed, resulting in the output "9".

A is decremented by 1, so A becomes 2.

Step 5: Iteration 4

A (2) is still greater than or equal to B (3).

The square of A (2²) is displayed, resulting in the output "4".

A is decremented by 1, so A becomes 1.

Step 6: Iteration 5

A (1) is still greater than or equal to B (3).

The square of A (1²) is displayed, resulting in the output "1".

A is decremented by 1, so A becomes 0.

Step 7: Loop termination

Since A (0) is no longer greater than or equal to B (3), the loop terminates.

Therefore, The output generated by the code execution will be "25, 16, 9, 4, 1" as the squares of A (starting from 5 and decreasing by 1) are displayed in each iteration of the loop.

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The length of a rectangle is six times its width.
If the perimeter of the rectangle is 70, find its length and width.

Answers

Answer:

L=25

W=10

Step-by-step explanation:

Answer:

Width = xLength = 6x

Use the perimeter to find x:

\(x + x + 6x + 6x = 70\\14x=70\\x=5\)

Therefore,

Width = x = 5Length = 6x = 6(5) = 30

Solve the given initial-value problem. the de is a bernoulli equation. y1/2 dy dx y3/2 = 1, y(0) = 9

Answers

A differential equation with some initial conditions is used to solve an initial value problem.

The required particular solution is given by the relation:

\($4y^{(3/2)} = 9e^{(-2x/3)} + 23\)

What is meant by an initial-value problem?

An initial value problem in multivariable calculus is an ordinary differential equation with an initial condition that specifies the value of the unknown function at a given point in the domain. In physics or other sciences, modeling a system frequently entails solving an initial value problem.

Let the given equation be \($y^{1/2} dy\ dx y^{3/2} = 1\), y(0) = 9

\($(\sqrt{y } ) y^{\prime}+\sqrt{(y^3\right\left) }=1\) …..(1)

Divide the given equation (1)  by \($\sqrt{ y} $\)  giving

\($y^{\prime}+y=y^{(-1 / 2)} \ldots(2)$\), which is in Bernoulli's form.

Put \($u=y^{(1+1 / 2)}=y^{(3 / 2)}$\)

Then \($(3 / 2) y^{(1 / 2)} \cdot y^{\prime}=u^{\prime}$\).

Multiply (2) by \($\sqrt{ } y$\) and we get

\(y^{(1 / 2)} y^{\prime}+y^{(3 / 2)}=1\)

(2/3) \(u^{\prime}+u=1$\) or \($u^{\prime}+(3 / 2) y=3 / 2$\),

which is a first order linear equation with an integrating factor

exp[Int{(2/3)dx}] = exp(2x/3) and a general solution is

\(u. $e^{(2 x / 3)}=(3 / 2) \ln \[\left[e^{(2 x / 3)} d x\right]+c\right.$\)  or

\(\mathrm{y}^{(3 / 2)} \cdot \mathrm{e}^{(2x / 3)}=(9 / 4) \mathrm{e}^{(2x / 3)}+{c}\)

To obtain the particular solution satisfying y(0) = 4,

put x = 0, y = 4, then

8 = (9/4) + c

c = (23/4)

Hence, the required particular solution is given by the relation:

\($4y^{(3/2)} = 9e^{(-2x/3)} + 23\)

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all you need is on the photo

please don't do step by step
pleaseeeeeeeeeeeeeeeeeeeeeeeeeeee​

all you need is on the photo please don't do step by step pleaseeeeeeeeeeeeeeeeeeeeeeeeeeee

Answers

Answer:

x1 = 6

x2 = -3

Step-by-step explanation:

you said not to :(

A consumer testing agency obtains four cars from each of six makes: Ford, Chevrolet, Nissan, Lincoln, Cadillac, and Mercedes. Makes 3 and 6 are imported while the others are domestic; makes 4, 5, and 6 are expensive while 1, 2, and 3 are less expensive; 1 and 4 are Ford products, while 2 and 5 are GM products. We wish to compare the six makes on their oil use per 100,000 miles driven. The mean responses by make of car were 4.6, 4.3, 4.4, 4.7, 4.8, and 6.2, and the sum of squares for error was 2.25.


Required:

a. Compute the Analysis of Variance table for this experiment. What would you conclude?

b. Design a set of contrasts that seem meaningful for this experiment and outline its purpose.

c. For each contrast, what conclusions can you draw using the Scheffé’s procedures?

Answers

To analyze the data from the consumer testing agency's experiment comparing oil use per 100,000 miles driven for six car makes, we can perform an Analysis of Variance (ANOVA).

The total sum of squares (SST) is calculated by subtracting the mean of all observations from each individual observation, squaring the differences, and summing them up. The sum of squares for error (SSE) is provided as 2.25. The total degrees of freedom (DF Total) is the total number of observations minus 1. The degrees of freedom for error (DF Error) is the total degrees of freedom minus the degrees of freedom for treatment (DF Treatment). The mean squares (MS) for treatment and error are calculated by dividing the sums of squares by their respective degrees of freedom. Finally, the F-ratio is obtained by dividing the MS for treatment by the MS for error.

The ANOVA table for this experiment would look as follows:

Source      |    Sum of Squares   | Degrees of Freedom | Mean Squares |    F-ratio

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

Treatment   |          ?                |          ?                   |        ?           |        ?

Error          |         2.25             |          ?                   |        ?           |

Total           |         ?                 |          ?                   |        -            |

To complete the ANOVA table and draw conclusions, we need additional information on the sums of squares, degrees of freedom, and mean squares for treatment. Without this information, we cannot determine the F-ratio or make definitive conclusions.

b. Contrasts are used to compare specific groups or combinations of groups within the data. In this experiment, meaningful contrasts could be:

1. Domestic vs. Imported: This contrast compares the average oil use of domestic cars (makes 1, 2, 4, and 5) with imported cars (makes 3 and 6).

2. Expensive vs. Less Expensive: This contrast compares the average oil use of expensive cars (makes 4, 5, and 6) with less expensive cars (makes 1, 2, and 3).

3. Ford vs. GM: This contrast compares the average oil use of Ford cars (makes 1 and 4) with GM cars (makes 2 and 5).

c. To draw conclusions using Scheffé's procedures, we would need the critical values for the specific contrasts and the corresponding mean square error. Without this information, we cannot determine the significance of the contrasts or make specific conclusions.

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Pls please help me ASAP!!!

Pls please help me ASAP!!!

Answers

Answer:

z = 0

Step-by-step explanation:

3 (z + 7) = 21

3z + 21 = 21

3z = 0

z = 0

Reposting because I didn’t get an answer and I need one asap. fake answers/links will be reported.

Reposting because I didnt get an answer and I need one asap. fake answers/links will be reported.

Answers

Answer:

B

Step-by-step explanation:

(x+5)^2

x^2+10x+25

(y+3)^2

y^2+6y+9

then it would be

x^2+10x+34+y^2+6y

subtract 34-36 since ur moving 36 to the left

where u get -2

so ur final equation would be x^2+y^2+10x+6y-2=0

Write the equations of the following ellipes in their colonical forms and hence determine the
a] Their Co-Ordinates of their ellispes
b] Their area of the ellipses
c] Their perimeter of the ellipse
d] Their vertices
e] Their foci
f ] Length of major and minor axis
The equation of ellipse are 4x² + 5y ² - 24x² - 20y + 36= 0 2x² ‐ 5y² + 8x + 10y + 13= 0​

Answers

a) The coordinates of the ellipse are centered at (3, 2).

b) The area of the ellipse is a = √5 and b = √20.

c) Perimeter ≈ π * (3(a + b) - √((3a + b)(a + 3b))).

d) The vertices of the ellipse are located at (3 ± √5, 2) and (3, 2 ± √20).

e) The foci of the ellipse cannot be determined in this case because the equation does not contain information about the foci.

f) The length of the major axis is 2a, The length of the minor axis is 2b.

Let's analyze each given equation of the ellipse and determine the requested information.

a) Equation: 4x² + 5y² - 24x - 20y + 36 = 0

To write the equation in standard form, we need to complete the squares for both x and y terms.

Rearranging the terms:

4x² - 24x + 5y² - 20y + 36 = 0

Completing the squares for x:

4(x² - 6x) + 5y² - 20y + 36 = 0

4(x² - 6x + 9) + 5y² - 20y + 36 = 4(9)

4(x - 3)² + 5y² - 20y + 36 = 36

4(x - 3)² + 5(y² - 4y) = 0

Completing the squares for y:

4(x - 3)² + 5(y² - 4y + 4) = 0 + 5(4)

4(x - 3)² + 5(y - 2)² = 20

Comparing this with the standard form of the ellipse equation:

[(x - h)²/a²] + [(y - k)²/b²] = 1

We can see that a² = 5, b² = 20, h = 3, and k = 2.

b) Area of the ellipse:

The area of the ellipse can be calculated using the formula: Area = π * a * b, where a and b are the semi-major and semi-minor axes, respectively.

In this case, a = √5 and b = √20.

So, the area of the ellipse is Area = π * √5 * √20 = π * 2 * √5.

c) Perimeter of the ellipse:

There is no simple formula to calculate the exact perimeter of an ellipse. However, an approximation formula can be used: Perimeter ≈ π * (3(a + b) - √((3a + b)(a + 3b))).

In this case, a = √5 and b = √20.

Plugging in the values, we can calculate the approximate perimeter of the ellipse.

d) Vertices:

The vertices of the ellipse can be determined using the formula:

Vertex on the x-axis: (h ± a, k)

Vertex on the y-axis: (h, k ± b)

In this case, the vertices will be (3 ± √5, 2) and (3, 2 ± √20).

e) Foci:

The foci of the ellipse can be determined using the formula:

Foci on the x-axis: (h ± c, k)

Foci on the y-axis: (h, k ± c)

where c = √(a² - b²) for a > b.

In this case, c = √(5 - 20) = √(-15) = imaginary value (since it is negative).

f) Length of major and minor axes:

The length of the major axis is 2a, where a = √5.

The length of the minor axis is 2b, where b = √20.

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Solve the proportion. 15. 7/11 = 18/x+1

Answers

7/11 = 18/x+1

7x + 7 = 198

7x = 198 - 7

7x = 191

x = 191/7

x = 27.28

Geometry, please answer question ASAP

Geometry, please answer question ASAP

Answers

9514 1404 393

Answer:

  C.  16

Step-by-step explanation:

UV is a diameter of the circle. If it is perpendicular to the chord XS, then it bisects that chord. This means ...

  XT = TS

  3b -1 = b +5

  2b = 6

  b = 3

  TS = b +5 = 8

The chord length is 2·TS = 2·8 = 16 units.

Answer:

C,XS=16

Step-by-step explanation:

3b-1=b+5

3b-b=5+1

2b=6

b=6/2=3

TS=b+5=3+5=8

XS=2 TS=2×8=16

Help me please it’s due today and I really need this

Help me please its due today and I really need this

Answers

Answer:

J

Step-by-step explanation:

If you look closely, you can see the slope line is decreasing which means all the positive slopes are automatically wrong, and second the first point the line intersects with the graph is 6 down which means the answer is -6

evaluate the integral. (use c for the constant of integration.) ∫ tan^3(x)sec^6(x)dxI am stuck on the first step.According to my book, it says "if the power of the secant is even, save a factor of sec^2(x) and use sec^2(x) = 1 + tan^2(x) to express the remaining factors in terms of tan(x)If the power of tangent is odd, save a factor of sec(x)tan(x) and use tan^2(x) = sec^2(x) - 1 to express the remaining factors in terms of sec(x).The power of tangent is odd and the power of secant is even, what should I do next? Can I apply both rules to the equation? Or will it be wrong? Please show steps and help guide me.

Answers

Answer:

\(\displaystyle {\frac{sec^8(x)}{8}-\frac{sec^6(x)}{6}+C}\)

Step-by-step explanation:

\(\displaystyle \int {\tan^3(x)\sec^6(x)} \, dx\\\\=\int {\tan^2(x)\sec^5(x)\tan(x)\sec(x)} \, dx\\=\int {\bigr(\sec^2(x)-1\bigr)\sec^5(x)\tan(x)\sec(x)} \, dx\\\)

Now let \(u=\sec(x)\) and \(du=\tan(x)\sec(x)dx\):

\(\displaystyle \int {\bigr(u^2-1\bigr)u^5} \, du\\\\=\int {u^7-u^5} \, du\\\\=\frac{u^8}{8}-\frac{u^6}{6}+C\\\\=\bold{\frac{sec^8(x)}{8}-\frac{sec^6(x)}{6}+C}\)

So, because the tangent term was odd and the secant term was even, we had to break it up so we could have the factor sec(x)tan(x), and then turn tan²(x) into sec²(x)-1. Also, because the derivative of sec(x) is sec(x)tan(x), the substitution becomes really simple as you see above.

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