leave your answer in simplified radical form.

Leave Your Answer In Simplified Radical Form.

Answers

Answer 1

Answer:

The .jpeg file is the answer. Others are formulas that I use to solve.

Leave Your Answer In Simplified Radical Form.
Leave Your Answer In Simplified Radical Form.
Leave Your Answer In Simplified Radical Form.

Related Questions

what is heavier ten tons of wool or ten tons of steel​

Answers

They weigh the same since they are both ten tons
both are equal as their weight is same

Given the formula A = 5h (B + b); solve for B.
2

Answers

Answer:

A=5h(B+b)

A/5h=B+b

A/5h - b= B

simplify 7-(3n+6)+10n​

Answers

Answer:

1 + 7n

Step-by-step explanation:

7-(3n+6)+10n​

7 - 3n - 6 + 10 n

1 - 7n

Answered by Gauthmath

using the unit circle what is the exact value of tanpi/6

Answers

Answer:

[tex] \frac{ \sqrt{3} }{3} [/tex]

Step-by-step explanation:

[tex]\frac{1}{3} \div \sqrt{3} [/tex]

Suppose that a category of world-class runners are known to run a marathon in an average of 147 minutes with a standard deviation of 12 minutes. Consider 49 of the races. Find the probability that the runner will average between 146 and 150 minutes in these 49 marathons. (Round your answer to two decimal places.)

Answers

Answer:

0.6524 = 65.24% probability that the runner will average between 146 and 150 minutes in these 49 marathons.

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal Probability Distribution

Problems of normal distributions can be solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

Average of 147 minutes with a standard deviation of 12 minutes.

This means that [tex]\mu = 147, \sigma = 12[/tex]

Consider 49 of the races.

This means that [tex]n = 49, s = \frac{12}{\sqrt{49}} = \frac{12}{7} = 1.7143[/tex]

Find the probability that the runner will average between 146 and 150 minutes in these 49 marathons.

This is the p-value of Z when X = 150 subtracted by the p-value of Z when X = 146. So

X = 150

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

By the Central Limit Theorem

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{150 - 147}{1.7143}[/tex]

[tex]Z = 1.75[/tex]

[tex]Z = 1.75[/tex] has a p-value of 0.9599

X = 146

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{146 - 147}{1.7143}[/tex]

[tex]Z = -0.583[/tex]

[tex]Z = -0.583[/tex] has a p-value of 0.3075.

0.9599 - 0.3075 = 0.6524.

0.6524 = 65.24% probability that the runner will average between 146 and 150 minutes in these 49 marathons.

Suppose the volume of the cone is 324pi Find dy/dx when x=6 and y=27

Answers

Answer:

[tex]\displaystyle \frac{dy}{dx} \bigg| \limits_{x = 6} = -9[/tex]

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to Right  

Equality Properties

Multiplication Property of Equality Division Property of Equality Addition Property of Equality Subtraction Property of Equality

Calculus

Differentiation

DerivativesDerivative Notation

Basic Power Rule:

f(x) = cxⁿf’(x) = c·nxⁿ⁻¹

Derivative Rule [Quotient Rule]:                                                                           [tex]\displaystyle \frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}[/tex]

Step-by-step explanation:

Step 1: Define

Identify

[tex]\displaystyle V = \frac{1}{3} \pi x^2y[/tex]

[tex]\displaystyle V = 324 \pi[/tex]

[tex]\displaystyle x = 6[/tex]

[tex]\displaystyle y = 27[/tex]

Step 2: Differentiate

Substitute in volume [Volume Formula]:                                                     [tex]\displaystyle 324 \pi = \frac{1}{3} \pi x^2y[/tex][Equality Properties] Rewrite:                                                                        [tex]\displaystyle y = \frac{972}{x^2}[/tex]Quotient Rule:                                                                                               [tex]\displaystyle \frac{dy}{dx} = \frac{(972)'x^2 - (x^2)'972}{(x^2)^2}[/tex]Basic Power Rule:                                                                                         [tex]\displaystyle \frac{dy}{dx} = \frac{0x^2 - (2x)972}{(x^2)^2}[/tex]Simplify:                                                                                                         [tex]\displaystyle \frac{dy}{dx} = \frac{-1944x}{x^4}[/tex]Simplify:                                                                                                         [tex]\displaystyle \frac{dy}{dx} = \frac{-1944}{x^3}[/tex]

Step 3: Evaluate

Substitute in variables [Derivative]:                                                             [tex]\displaystyle \frac{dy}{dx} \bigg| \limits_{x = 6} = \frac{-1944}{6^3}[/tex]Simplify:                                                                                                         [tex]\displaystyle \frac{dy}{dx} \bigg| \limits_{x = 6} = -9[/tex]

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Differentiation

15. Five boys went to see the CIRCUS. Four of them had Rs.5 each and the fifth boy had Re.1 more than the entrance ticket price. IF with the whole amount (which the 5 boys had), the boys were able to just buy the entrance ticket for all the 5, cost of the entrance ticket per person was​

Answers

Answer:

20+(x+1) = 5x

x=21/4

x= 5.25

The entrance ticket per person can be calculated using algebraic equation. We have create the algebraic expression as per the question.

The entrance ticket per person is Rs. 5.25.

Given:

Total boys are 5

4 boys has 5 rupee each so total rupee are [tex]=5\times 4=20[/tex].

Let the entrance ticket per boy is [tex]x[/tex].

One boy had 1 rupee more than entrance ticket [tex](x+1)[/tex].

Write the algebraic expression to calculate the entrance ticket per person.

[tex]5x=20+(x+1)\\5x=20+x+1\\5x-x=20+1\\4x=21\\x=5.25[/tex]

Thus, the entrance ticket per person is Rs. 5.25.

Learn more about algebraic expression here:

https://brainly.com/question/953809

Please help me solve the question...

Answers

Answer:

All of area is πr²

Step-by-step explanation:

and we should write the area kind of radian. So if all area is π*(12)²= 144π - this for 360°- and 240° is 96π

but we must add area of the triangle which has two same side and has 6 high so it's area is 6*12√3/2 = 36✓3

our answer is 96π+ 36✓3

Help me plz need the steps


In Figure 7 (open photo), GH is a diameter of the circle. What is
x² + y² ?

A)58
(B) 49
(C) 10
D) 9
(E) It cannot be determined from the information given.

Answers

Answer:

[tex]A)58[/tex]

Step-by-step explanation:

[tex][Kindly\ refer\ the\ attachment]\\We\ are\ given:\\GH\ is\ the\ diameter\ of\ the\ circle.\\ Also,\\ GM=3\ units\ and\ MH=7\ units\\From\ the\ figure,\\GH\ subtends\ an\ angle\ at\ two\ points\ specifically\ M\ and\ N\ on\ the\\ arc.\\Now,\\We\ know\ that,\\'The\ Angle\ subtended\ by\ the\ diameter\ anywhere\ over\ the\ arc\ of\ the\\ circle\ is\ always\ 90\ degrees'.\\Hence,\\\angle GMH\ = \angle GNH=90\\[/tex]

[tex]Also,\\Pythagoras\ Theorem\ states\ that: 'In\ a\ right\ triangle,\ the\ sum\ of\ squares\\ \ of\ the\ legs\ is\ equal\ to\ the\ square\ of\ the\ hypotenuse'\\In\ \triangle GMH,\\Since\ \angle GMH=90,\\GM^2+MH^2=GH^2[Through\ Pythagoras\ Theorem]\\Hence,\\Substituting\ GM=3,\ MH=7:\\3^2+7^2=GH^2\\GH^2=9+49=58[/tex]

[tex]Similarly,\\In\ \triangle GNH,\\Since\ \angle GNH=90,\\GN^2+NH^2=GH^2\\Hence,\\Substituting\ GN=x\ and\ NH=y:\\x^2+y^2=GH^2\\x^2+y^2=58[/tex]

I need help please. Thank you

Answers

Answer:

0.0009765625

Step-by-step explanation:

This is what i got its probally incorrect

Y=square root of x compare to y= - square root of x how they differ and why

Answers

Answer:

Simply because x=y2 doesn't imply that y=

x

.

The figure below is a rectangular prism.
Which edge is parallel to segment BD?
A. HK
B. BM
C. DK
D. AH

Answers

D. AH

Segment AH is parallel to BD and if the lines were drawn out infinitely, the two lines would never meet.

Plz answer asap question in picture

Answers

Answer:

-1 <x < 7

(-1,7)

Step-by-step explanation:

open circle on the left means the number is greater than

-1 <x

Open circle on the right means the number is less than

x < 7

Since both statements are true. we combine them

-1 <x < 7

open circles means parentheses, closed circles mean brackets

find the common ratio of the geometric sequence 4,3,9/4

Answers

Answer:

3/4

Step-by-step explanation:

To find the common ratio, take the second term and divide by the first term

3/4

Check with the third and second terms

9/4 ÷3

9/4 *1/3= 3/4

The common ratio is 3/4

Which of the following quadratic equations is written in general form?

Answers

The 3rd one

Step-by-step explanation:

Reason being that it has no factors, meaning it cannot be simplified any further, but quadratic method can be used to attain factors.

hope it makes sense :)

What is the measure of the angle in red?
70°

Answers

Answer:

See pic below.

Answer: C. 380°

Step-by-step explanation:

____________ is/are when data is analyzed in order to make decisions about the population behind the data.
A. Experiments
B. Simulations
C. Surveys
D. Statistics

Answers

Answer:

statistics are when data analyzed in order to make decisions about the population behind the data.

The correct answer to the question is Statistics (Option D).

What is statistics?

Statistics is a field of study in mathematics which deals with raw data. It is a tool which refines the data to produce meaningful results and a pathway to better understanding of it.

Some examples of raw data can be population sample which loves to see a particular TV show or a sample of students getting so and so marks in Math test.

Data is analyzed mainly by three different measure of central tendency that is mean, median and mode.

Mean is the average value of given discrete data.Median is the middle value when the data is sorted in ascending or descending order.Mode is the value that has highest frequency.

Therefore, Statistics the correct answer.

To know more about statistics refer:https://brainly.com/question/10734660

#SPJ2

PLEASE HELP THIS IS DUE ASAP (answer in decimal!!!!)

Answers

Your answer should be 0.14

The coordinates of the preimage are:

A(−8,−2)

B(−4,−3)

C(−2,−8)

D(−10,−6)

Now let’s find the coordinates after the reflection over the x-axis.

A′(−8,
)

B′(−4,
)

C′(−2,
)

D′(−10,
)

And now find the coordinates after the reflection over the y-axis.

A′′(
,2)

B′′(
,3)

C′′(
,8)

D′′(
,6)

This is also the same as a rotation of 180∘.

Answers

9514 1404 393

Answer:

A'(-8, 2) ⇒ A"(8, 2)B'(-4, 3) ⇒ B"(4, 3)C'(-2, 8) ⇒ C"(2, 8)D'(-10, 6) ⇒ D"(10, 6)

Step-by-step explanation:

Reflection over the x-axis changes the sign of the y-coordinate. Reflection over the y-axis changes the sign of the x-coordinate. We can summarize the transformations as ...

  preimage point ⇒ reflection over x ⇒ reflection over y

A(−8,−2) ⇒ A'(-8, 2) ⇒ A"(8, 2)

B(−4,−3) ⇒ B'(-4, 3) ⇒ B"(4, 3)

C(−2,−8) ⇒ C'(-2, 8) ⇒ C"(2, 8)

D(−10,−6) ⇒ D'(-10, 6) ⇒ D"(10, 6)

The sum of two numbers is 41. The larger number is 17 more than the smaller number. What are the numbers?
Larger number:
Smaller number:

Answers

Let x = the larger number
Let x - 17 = the smaller number
x + x + 17 = 41
2x + 17 = 41
2x = 58
x = 29
x - 17 = 29 - 17 = 12

What is 1.25 x 10^8 in standard form?

Answers

Answer:

125000000

Step-by-step explanation:

1.25 x 10^8

Move the decimal 8 places to the right

1.25

We can move it two places

125

We need to add 6 more zeros

125000000

Answer: 125,000,000

Step-by-step explanation:

The differential equation of a certain system is 20⁢y′⁢′+c⁢y′+80⁢y=0
, where c is called damping constant for what value of c critical damping hapens

Options:

110

64

50

60

Answers

Answer:

c=80

Step-by-step explanation:

Based on my reading the critical damping occurs when the discriminant of the quadratic characteristic equation is 0.

So let's see that characteristic equation:

20⁢r^2+c⁢r+80⁢=0

The discriminant can be found by calculating b^2-4aC of ar^2+br+C=0.

a=20

b=c

C=80

c^2-4(20)(80)

We want this to be 0.

c^2-4(20)(80)=0

Simplify:

c^2-6400=0

Add 6400 on both sides:

c^2=6400

Take square root of both sides:

c=80 or c=-80

Based on further reading damping equations in form

a⁢y′⁢′+b⁢y′+C⁢y=0

should have positive coefficients with b also having the possibility of being zero.

Armando's carpet has an area of 220 square feet. He hires the Carpet Pro carpet cleaning service, which is able to clean 9 square feet of his carpet every minute. Therefore, in the first minute, CarpetPro is able to clean 9 square feet of Armando's carpet (leaving 211 square feet remaining to be cleaned). In the second minute, the cleaner is again able to clean 9 square feet of carpet (leaving 202 square feet to be cleaned), etc. Which of the following functions expresses the number of square feet that still remain to be cleaned after t minutes from the time that the carpet cleaner began cleaning this carpet.
a. A(t) = 220 - 9
b. A(t) = 220 - 0.1
c. A(t) = 220(0.1)
d. A(t) = 220(0.96)
e. A(t) 220 - 0.96

Answers

Answer:

The answer is A

Step-by-step explanation:

The answer needs to be in slope-intercept form or y=mx+b because the amount of carpet being cleaned every minute is a constant decrease of 9 square feet.

So our b value (or how much carpet cleaned at 0 minutes) is 220

And our m value (or slope or how much carpet is being cleaned every minute) is 9

So if we plug the variables with the tangible numbers we get A(t) = -9x+220 or A(t) = 220-9x.

provided by gauth math

The sum of 'n' terms of an arithmetic sequence is 4n^2+3n. What is the first term, the common difference, and the sequence?

Answers

Answer:

The sequence has first term 7 and common difference is 8.

So the sequence is f(n)=7 + 8(n-1)

Step-by-step explanation:

Let a be the first term.

Let a+d be the second term where d is the common difference.

Then a+2d is the third....

And a+(n-1)d is the nth term.

Adding these terms we get:

an+(n-1)(n)/2×d

For the first term of this sum I seen we had n amount of a's and for the second term I used the well known identity sum of the first n positive integers is n(n+1)/2.

Let's simplify:

an+(n-1)(n)/2×d

Distribute:

an+(n^2d/2)-(nd/2)

Find common denominator:

(2an/2)+(n^2d/2)-(nd/2)

Combine terms into one:

(2an+n^2d-nd)/2

Reorder terms:

(n^2d+2an-nd)/2

Regroup terms:

(n^2d+(2a-d)n)/2

We want the following sum though:

4n^2+3n

This means d/2=4 (so d=8) and (2a-d)/2=3.

So plug d=8 into second equation to solve for a.

(2a-8)/2=3

2a-8=6

2a=14

a=7

The sequence has first term 7 and common difference is 8.

So the sequence is f(n)=7 + 8(n-1).

help please i’ll give brainliest

Answers

Answer:

3rd option

..................

[tex]A) \frac{4-2}{2-1} =2[/tex]

[tex]B)\frac{1-0.5}{4-2} =0.25[/tex]

[tex]C)>2[/tex]

[tex]D) 1[/tex]

~OAmalOHopeO

Answer:

The third one is your answer

What is the dimension of the vector space consisting of five-by-one column matrices where the rows sum to zero and the first row is equal to the second row?
a. 5
b. 4
c. 3
d. 2

Answers

Answer:

Option c.

Step-by-step explanation:

If we have a vector of N components (or variables), and we have K linear independent restrictions for these N components (such that K < N, we can't have more restrictions than components.)

The dimension of the vector will be given by N - K.

Here we know that we have a vector of 5 components, that can be written as:

[tex]v = \left[\begin{array}{ccc}v_1\\v_2\\v_3\\v_4\\v_5\end{array}\right][/tex]

And we have two restrictions, so we can expect that the dimension of the vector is:

5 - 2 = 3

But let's see it, the restrictions are:

"the first row is equal to the second row"

Then we can rewrite our vector as:

[tex]v = \left[\begin{array}{ccc}v_1\\v_1\\v_3\\v_4\\v_5\end{array}\right][/tex]

Notice that now we have only 4 variables, v₁, v₃, v₄, and v₅

We also know that the sum of the rows is equal to zero, thus:

v₁ + v₂ + v₃ + v₄ + v₅ = 0

we know that v₂ = v₁, so we can replace that to get:

2*v₁ + v₃ + v₄ + v₅ = 0

Now we can isolate one of the variables, to write it in term of the others, for example, let's isolate v₅:

v₅ = -2*v₁ - v₃ - v₄

Now if we replace that in our vector, we have:

[tex]v = \left[\begin{array}{ccc}v_1\\v_1\\v_3\\v_4\\-2*v_1 - v_3 - v_4\end{array}\right][/tex]

Notice that our vector depends on only 3 variables,  v₁, v₃, and v₄, so we can define our vector in a 3-dimensional space.

Then the correct option is c, the dimension of the vector space is 3.

Put the following equation of a line into slope-intercept form, simplifying all fractions 2x-2y=14

Answers

Answer:

y = x - 7

Step-by-step explanation:

Answer: y = x - 7

Slope intercept form: y = mx + b

[tex]2x-2y=14\\\\-2y=-2x+14\\\\y=\frac{-2x+14}{-2} =\frac{-2(x-7)}{-2} =x-7[/tex]

1. Ten times the sum of -270 and a number gives -20.​

Answers

9514 1404 393

Answer:

equation: 10(-270 +n) = -20number: 268

Step-by-step explanation:

If n represents the number, we have ...

  10(-270 +n) = -20 . . . an equation for n

__

The solution can be found as ...

  -270 +n = -2 . . . . . divide by 10

  n = 268 . . . . . . . add 270

The number is 268.

Susan has an investment account which compounds interest annually at a rate of 3.2%. After 6 years, she has 86125 in the
account. How much money did she initially place in the account? Round your answer to the nearest whole number. Do not
include a s in your answer.
Provide your answer below:

Answers

Answer:

10610

Step-by-step explanation:

Given,

T=6years

R=3.2%

A=86125

Now,

CA=P[1+R/10]^T

or,P=86125/[1+3.2/10]^6

=86125/5.29

=10610

A bag contains 8 red balls and 3 white balls. Two balls are drawn without replacement. (Enter your probabilities as fractions.) (a) What is the probability that the second ball is white, given that the first ball is red

Answers

Answer:

12/55

Step-by-step explanation:

Probability is the ratio of the number of possible outcomes to the number of total outcomes.

Given that the bag contains 8 red balls and 3 white balls, the probability of picking a red ball

p(r) = 8/(8+3) = 8/11

Probability of picking a white ball

= 3/11

when a red ball is picked first, the total number of balls reduces to 10 hence the probability that the second ball is white, given that the first ball is red

=8/11 * 3/10

= 24/110

= 12/55

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BRAINLIEST graphing question Nitric acid can be formed in two steps from the atmospheric gases nitrogen and oxygen, plus hydrogen prepared by reforming natural gas. In the first step, nitrogen and hydrogen react to form ammonia: N2 (g) + 3H2 (g) 2NH3 (g) =H92.kJ In the second step, ammonia and oxygen react to form nitric acid and water: NH3 (g) + 2O2 (g) HNO3 (g) + H2O (g) =H330.kJ Required:Calculate the net change in enthalpy for the formation of one mole of nitric acid from nitrogen, hydrogen and oxygen from these reactions. A runner measured the time it took him to run 5 km. What can he calculate from the distance and time measurements? *A directionB massC speedD force 1.You will meet the tour guide on your at the hotel. (ARRIVE)2. She felt that she passed the exam. (RELIEF)3. Could you give me a ...., please? (BAND)4. Huongs listening skill is and she really needs to improve it. (SATISFY)5. After many years of , Edison finally succeeded in inventing the light bulb. (FAIL)6. She owes her ..largely to her sense of humor. (POPULAR)7. Joe and Kate want to visit an ..institute. (OCEAN)8. Giang put .on that towel to stop the bleeding. (PRESS)9. The lovers stood, hand in hand, gazing at the ..sky. (STAR)10. The new film is .....good. (EXCEPTION what tone the author invoke when he writes "only a woman could have believed it? QUESTION 11 1. Through PCR, we have determined the PER3 genotypes for a class of students as follows: H4/H4 = 125 individuals; H4/H5 = 85 individuals and H5/H5=24 individuals. Using the allele's frequencies experimentally derived, calculate the frequency of the H4/H5 genotype that would be expected if the class were a population in Hardy-Weinberg equilibrium. 1. 0.28 2. 0.51 3. 0.19 4. 0.72 5. 0.14 6. 0.24 7. 0.41 QUESTION 12 1. Through PCR, we have determined the PER3 genotypes for a class of students as follows: H4/H4 = 125 individuals; H4/H5 = 85 individuals and H5/H5=24 individuals. Using the genotype frequencies derived assuming that the class were a population in Hardy-Weinberg equilibrium, calculate the number of H4/H4 individuals that would be expected in the class (rounded numbers). 1. 19 2. 57 3. 72 4. 147 5. 171 6. 120 7. 96 QUESTION 13 1. Through PCR, we have determined the PER3 genotypes for a class of students as follows: H4/H4 = 125 individuals; H4/H5 = 85 individuals and H5/H5=24 individuals. Considering the Hardy Weinberg equilibrium and comparing the observed and the expected number of individuals for the three genotypes, calculate the value of the Chi-square statistic 1. 2.69 2. 0.05 3. 28.67 4. 14.59 5. 0.50 6. 22.31 7. 3.84 QUESTION 14 1. Through PCR, we have determined the PER3 genotypes for a class of students as follows: H4/H4 = 125 individuals; H4/H5 = 85 individuals and H5/H5=24 individuals. Considering the Hardy Weinberg equilibrium, which is the (correct) null hypothesis tested by Chi-square? 1. The whole class represents a population that is in Hardy-Weinberg equilibrium 2. The whole class represents a population that may not be in Hardy-Weinberg equilibrium 3. The whole class represents a population that is not in Hardy-Weinberg equilibrium 4. The whole class represents a population that may be in Hardy-Weinberg equilibrium QUESTION 15 1. Through PCR, we have determined the PER3 genotypes for a class of students as follows: H4/H4 = 125 individuals; H4/H5 = 85 individuals and H5/H5=24 individuals. Considering the Hardy Weinberg equilibrium and calculating the Chi-square statistic, do you reject or fail to reject the null-hypothesis? 1. Comparing the value of Chi-square that I calculated with the critical value of Chi-square for the appropriate degrees of freedom I conclude that P>0.05. Hence, I reject the null hypothesis. 2. Comparing the value of Chi-square that I calculated with the critical value of Chi-square for the appropriate degrees of freedom, I conclude that P>0.05. Hence, I fail to reject the null hypothesis. 3. Comparing the value of Chi-square that I calculated with the critical value of Chi-square for the appropriate degrees of freedom I conclude that P Did you enjoy the flight? (both)