How is a constant term different than a variable for an expression that represents a real word situation

The integration by part  is [tex]\frac{x^{3}In x }{3} -\frac{x^{3} }{9} +C[/tex] with indicated choices of u and dv.

The method Integration by Parts is known to be a special method of integration that is often useful. In calculus, definite integrals are referred to as the integral with limits such as upper and lower limits. It is also possible to derive the formula of integration by parts with limits.

Given: ∫ [tex]x^{2} In x dx[/tex] ; [tex]u=In x[/tex] ; [tex]dv=x^{2} dx[/tex]

Consider  [tex]u=In x[/tex] and [tex]dv=x^{2} dx[/tex]

⇒ [tex]u=In x[/tex]             ⇒  [tex]dv=x^{2} dx[/tex]

⇒ [tex]du= \frac{1}{x} dx[/tex]           ⇒ [tex]v=\frac{x^{3} }{3}[/tex]

Now, apply integration by parts to solve the integral

∫ u dv =  uv - ∫ v du

the antiderivative of [tex]x^{n}[/tex][tex]=\frac{x^{n+1} }{n+1}[/tex] and the derivative of In x = [tex]\frac{1}{x}[/tex] .

⇒ ∫ [tex]x^{2} In x dx =[/tex][tex]\frac{x^{3} In x}{3} -\frac{x^{3} }{3} .\frac{1}{x} dx[/tex]

                     = [tex]\frac{x^{3}In x }{3}[/tex] [tex]-\frac{x^{2} }{3} dx[/tex]

                     =  [tex]\frac{x^{3}In x }{3}[/tex] [tex]-\frac{x^{3} }{9} +C[/tex]

Hence, the final integration is   [tex]\frac{x^{3}In x }{3}[/tex] [tex]-\frac{x^{3} }{9} +C[/tex]

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The triangles in the diagram indicates that ΔABD is an isosceles triangle, therefore;∠2 ≅ ∠1

The two-column method can be used to prove that the angle m∠BAC > m∠C as follows;

Statement       [tex]{}[/tex]                         Reasons

1. BC > BA       [tex]{}[/tex]                         1. Given

2. BA ≅ BD      [tex]{}[/tex]                        2. Given

3. ΔABD is an isosceles Δ[tex]{}[/tex]      3. Definition of isosceles triangle

4. ∠1 ≅ ∠2      [tex]{}[/tex]                         4. Base ∠s of isosceles triangle ΔABD

5. ∠1 = ∠2      [tex]{}[/tex]                          5. Definition of congruency

6. ∠1 = ∠3 + ∠C      [tex]{}[/tex]                 6. Exterior angle theorem

7. ∠2 = ∠3 + ∠C      [tex]{}[/tex]                7. Substitution property

8. ∠2 - ∠3 = ∠C [tex]{}[/tex]                     8. Subtraction property

9. ∠2 > ∠C [tex]{}[/tex]                             9. Definition

An isosceles triangle is a triangle that have a pair of congruent sides and one pair of congruent angles.

The reasons in the above two-columns used to prove m∠BAC > m∠C are as follows;

Isosceles triangle ΔABD

The triangle ΔABD that have side AB congruent to side BD is an isosceles triangle by definition of an isosceles triangle.

The base angles of an isosceles triangle that are angles opposite congruent sides in a triangle are also congruent.

Definition of congruency

A geometric figure is congruent to another figure if their measurements are the same.

Exterior angle theorem

The exterior angle of a triangle theorem states that the exterior angle of a triangle is equal to the sum of the two interior opposite angles.

Substitution property

The substitution property of equality states that if a quantity a is equal to another quantity b then a can be substituted by b in expressions and equations without changing their value.

Subtraction property

The subtraction property of equality states that if the same quantity is subtracted from both sides of an equation, then the new amount obtained are also equal

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AS per the permutation concept, the number of ways can Alice win or draw exactly three times is 240

Permutation:

In probability, permutation refers the arrangement of objects in a definite order.

Given,

Alice and Bob are playing a game in which there are several rounds and each round can be a win, a loss, or a draw for Alice.

Here we need to find the number of ways can Alice win or draw exactly three times, if they play 6 rounds.

While we looking into the given question, we know that,

Total number of games = 6

Possible values = 3 ( win, loss, or draw)

Here we need to find the probability of winning 3 times, it can be calculated by using the permutation as,

=> 6! / (6 - 3)!

=> 6! / 3!

=> 120

Similarly, the the probability of draw 3 times, it can be calculated by using the permutation as,

=> 6!/ (6 - 3)!

=> 120

So, the number of ways can Alice win or draw exactly three times is

=> 120 + 120 =  240

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Answer:

Step-by-step explanation:

.36

The margin of error for the confidence interval (6.5, 6.9) is 0.2 , the correct option is (c) .

In the question ,

it is given that ,

the 95% confidence interval for mean hours slept in the graduate students is (6.5, 6.9) ,

we have to find the margin of error ,

we know that the formula for the margin of error is

= (upper limit of confidence interval - lower limit )/2

From the given interval ,

we see that ,

upper limit of confidence interval = 6.9

lower limit of confidence interval = 6.5

Substituting the above value in margin of error formula ,

we get ,

margin of error = (6.9 - 6.5)/2

= 0.4/2

= 0.2

Therefore , the margin of error is 0.2   .

The given question is incomplete , the complete question is

Suppose that based on a random sample, a 95% confidence interval for the mean hours slept (per day) among graduate students was found to be (6.5, 6.9). What is the margin of error of this confidence interval? responses

(a) 6.7

(b) 0.4

(c) 0.2

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The intervals on which the function f(x) = ax² + bx + c (where "a" doesn't = 0) is increasing and decreasing is x>-b/2a and x<-b/2a

Given that,

We have to find the intervals on which the function f(x) = ax² + bx + c (where "a" doesn't = 0) is increasing and decreasing.

We know that,

Take the function equation,

f(x)=ax² + bx + c

The derivative of the function of x is  

f'(x)=ax+b

So, f(x) is increasing when f'(x) > 0

f(x) is decreasing when f'(x) < 0

Then we can say

f'(x) > 0 , when  b > 0  and a < 0

2ax + b < 0

2ax < - b

x<-b/2a

Now,

f'(x) < 0 , when  b < 0  and a > 0

2ax + b > 0

2ax > - b

x>-b/2a

Therefore, the intervals on which the function f(x) = ax² + bx + c (where "a" doesn't = 0) is increasing and decreasing is x>-b/2a and x<-b/2a.

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Using mathematical operations we can conclude that the rate of lemons being used in a cup of lemonade is 1/4.

Calculating a value using operands and a math operator is referred to as performing a mathematical "operation."

The math operator's symbol has predetermined rules that must be applied to the supplied operands or numbers.

In mathematics, addition, subtraction, multiplication, division, and modular forms are the five basic operations.

So, we know that:

Lemons required to make 16 cups of lemonade: 4 lemons

Then, the rate of lemons being used in lemonade per cup:

4/16

1/4

Therefore, using mathematical operations we can conclude that the rate of lemons being used in a cup of lemonade is 1/4.

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a) The ratio of tomato plants to lettuce plants:

b) The ratio of lettuce plants to cucumber plants:

a) The expected return for stocks a and b is 13.56% and 20.02% respectively.

b) The standard deviation for stocks a and b are 8.32% and 32.60%.

a) Expected return = Probability of recession × Return during recession + Probability of normal × Return during normal + Probability of boom × Return during boom

Expected return for stock A = 0.21 × 0.04 + 0.61 × 0.12 + 0.18 × 0.30

The expected return for stock A = 0.1356

The expected return for stock A = 13.56%

Expected return for stock B = 0.21 × (-0.41) + 0.61 × 0.31 + 0.18 × 0.54

The expected return for stock B = 0.2002

The expected return for stock B = 20.02%

b) Standard deviation of stock = √{Probability(Recession) × (Rate during recession - expected rate )² + Probability(Normal)(Rate during normal - expected return)² + Probability(Boom) × (Rate in boom - Expected return)²}

Standard deviation of stock A = √[(0.21 × (0.04-0.1356)² + 0.61 × (0.12 - 0.1356)² + 0.18 × (0.30-0.1356)²)]

The standard deviation of stock A = 0.0832

The standard deviation of stock A = 8.32

Standard deviation of stock B = √(0.21 × (-0.41-0.2002)² + 0.61(0.31 - 0.2002)² + 0.18 × (0.54-0.2002)²)]

The standard deviation of stock B = 0.3260

The standard deviation of stock B = 32.60%

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Answer:

$200

Step-by-step explanation:

so if peter has been depositing 50 each month for 8 months then you would just do 50x8 which equals 400 and 600-400=200

To guarantee that at least on pair o the numbers adds up to 9, you must select at least 3 numbers from the set by combination

1. First, select any two numbers from the set.

2. Now, calculate their sum.

3. If the sum of the two numbers is not equal to 9, select a third number from the set.

4. Determine the three numbers' sum.

5. If the sum of the three numbers is still not equal to 9, select a fourth number from the set and calculate the sum.

6. Repeat this process until the sum of the selected numbers is equal to 9 by combination

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Answer:

Using an Equation

Simplify the equation as closely as possible to the form of y = mx + b. Check to see if your equation has exponents. If it has exponents, it is nonlinear. If your equation has no exponents, it is linear.

Answer:

Step-by-step explanation:

To be linear it must have the form  y=mx+b.IN a linear equation the variables are shown in first degree and terms with products of variables are absent .

Answer:

base = 10

Step-by-step explanation:

Answer:

None of the Above?

Step-by-step explanation:

1.) If we put this into words, we would get -5y+7/3y=-5-8/3y-4

2.) By moving all y terms to one side, we get -5y + 7/3y + 8/3y = -5 - 4

3.) By simplifying, you would get -4 2/3y = -9.
4.) If you divide both sides by -4 2/3, you would get that y = -9 * - 3/14 = 27/14

Answer:

XV = 16

Step-by-step explanation:

Triangle VWX is an isosceles triangle. Since the base angles are the same, the sides opposite the angle are also the same. See image. They gave VW so XV is the same as VW.

For dimensions of x =10 and y=5 for the base and height that will make the tank weight as little as possible.

Here we are given the information of a square-based, open-top, rectangular steel holding tank, so here the tank is in the shape of a cuboid and we know  the formula  for :

volume = x*y*z, as  here it is a square base, so the volume will be

volume =  x²*y, where x and y are the dimension of the base and height

So for the  given reason,  

=>x²*y =  500

=> y = 500/x²

Again, the surface area formula  will be

S=x²+4xy, substituting the value of  y in this question

=>S=x²+4x(500/x²)

=>S=x²+2000/x

Differentiating the above formula with respect to x, we get

=>2x-2000/x²=0

=>2x³= 2000

=>x³= 1000

=>x =10

so the substituting the x values of x in  the equation y , we get

y = 500/x²

=>y = 5

so the dimensions are x= 10 and y=5

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f(2) = g(2) and f(0) = g(0) is representation of f(x) = g(x)

Graph is a mathematical representation of a network and it describes the relationship between lines and points.

the curved red line represents g(x) and the straight blue line represents f(x) .

The important thing here is that the equality of functions f(x)=g(x) is represented as a common function between their curves. So, we just need to find such a common point for both.

f(x) has points (0, 4) and (2, 0).

g(x) has points (0,4) and (2,0).

Notice that both functions have the same y-value for x=0 and x=2, that means f(0)=g(0) and f(2)=g(2)

Hence, f(2) = g(2) and f(0) = g(0) is representation of f(x) = g(x)

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20 minutes is [tex]1/3^{rd}[/tex] of an hour

Answer:

y = [tex]\frac{3}{4}[/tex]x + 2

Step-by-step explanation:

To find the slope we will use the slope formula of y2-y1/x2-x1.

Our two y values are 2 and -1.

Our two x values are 0 and 4.

2 - (-1) = 3

0 - 4 = -4

3/4 = [tex]\frac{3}{4}[/tex]

y = [tex]\frac{3}{4}[/tex]x + 2

To check whether the true average is 59k or not, A sample mean T-Test needs to be conducted. The true value of 59k is 44k after the test.

In the field of statistics, average means that the ratio of the sum of the numbers of a given set, to the total number of characters in a given set. It is also called as the Arithmetic mean.

It is given that the student center has published the data that the average salary of the college graduates is 59k. Accordingly, the sample of 49 college students were verified and the sample average salary came out to be 44k.

When we analyze the  given question, we find that,

The total number of students were 49,The true average was found to be 59k,and the Sample average is found to be 44k.

Thus, in  the present question we will use the One- sample T-test.

We used this test, as the statistical hypothesis test is used to find if the unrecognized population mean is different from the specific value.

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The correct answer is (C) since the limitations to depict this real-world problem in inequality are 75 < x < 110 and 0 < y ≤ 50.

The equation-like form of the formula 5x 4 > 2x + 3 has an arrowhead in place of the equals sign. It is an illustration of inequity.

This shows that the left half, 5x 4, is bigger than the right part, 2x + 3.

So, our senior trip will be a group excursion to a beach.

Only when the temperature is between 75 and 110 degrees does the resort permit swimming.

On your excursion, there is space for 50 people.

Let y represent the number of travelers and x represent the temperature.

The temperature ranges from 75 to 110 degrees:

75 < x < 110  

On your excursion, there is space for 50 persons:

0 < y ≤ 50

Therefore, the correct answer is (C) since the limitations to depict this real-world problem in inequality are 75 < x < 110 and 0 < y ≤ 50.

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Complete question:
A class trip to a beach has been planned for your senior trip. The resort only allows swimming when the temperature is between 75 degrees and 110 degrees. There is room for 50 people on your trip. Write the constraints to represent this real-world problem, where x is the temperature and y is the number of people on your trip.

a. 0 < x ≤ 50 and 75 < y < 110

b. x > 75 and y < 110

c. 75 < x < 110 and 0 < y ≤ 50

d. x < 110 and y > 75

Answer:

I got you!

a ≤ 11

For more info, look at the steps in the attached image!

Step-by-step explanation:

Answer:

a ≥11

Step-by-step explanation:

[tex]21 \geqslant 3a - 21 + 9 \\ 21 \geqslant 3a - 12 \\ 21 + 12 \geqslant 3a - 12 + 12 \\ 33 \geqslant 3a \\ 3a \leqslant 33 \\ a \geqslant 11[/tex]

The range function to add up the series is 15 ,35 ,60 ,90, 125,165,210 ,260

A unit of code with a name is referred to as a function in Python. We utilize functions anytime we need to repeat a process repeatedly without writing the same code.

Given: Series 15, 20, 25, 30,.. 50

Program (Pascal ABC) and Result:

Program:

Var i, Summ : integer;

begin

i : = 15;

Summ : = 15;

While i <= 50 do

begin

WriteIn ( '   ' , Summ)  ;

i : = i + 5 ;

Summ : = Summ + i ;

end ;

end.

Result : 15 ,35 ,60 ,90, 125,165,210 ,260

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The relationship between the number of games Carlos buys and the total price is proportional.

A proportional relationship exists between two or more variables if they move in the same direction. If one of the variables increases, the other variable increases.

As the number of games that are bought increases, the total price paid also increases. This means that the relationship is proportional.

The formula that is used to represent proportional relationship is:

T = kg

Where:

T = $36g

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Answer:

whats the question

Answer:

What are you asking?

The final value of the product = 292

According to the given equation -

= (4a2+3b)x(4a2+36)

multiply the whole equation-

= 16a4+144+12a2b+108

as we can see that the values of a and b are given 1 and 2 respectively,

so putting the values of a and b in the equation we get -

= 16(1)4+144+12(1)2(2)+108

= 292

The product of the following equation will be = 292

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The nth term of the pattern is (1+n)/(18+n)

What is a sequence?

succession is the act of doing something after another. successional order: a list of books organized by title. a running or interconnected sequence: a series of sonnets. something that happens after; a following event; a result; or a repercussion. In contrast to a set, the order of the components in a sequence does important, and the same elements may occur more than once at various positions. A series can officially be described as a function from natural numbers.

We are given a sequence

1/18, 2/19, 3/20, 4/21

As we can see the second term is (1+1)/(18+1)

Third term is (1+2)/(18+2)

Hence the n th term of the sequence can be given as (1+n)/(18+n)

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