Evaluate the following integral: ∫20(45t3−34t2+25t)dt

Answer:

The given function is:

f(x) = 1/(x - 2), x ≠ 2

0, x = 2

To find the inverse of the function, we need to solve for x in terms of f(x).

Let y = f(x)

Case 1: When y = 0

If y = 0, then x = 2 as per the definition of the function.

Case 2: When y ≠ 0

y = 1/(x - 2)

1/y = x - 2

x = 1/y + 2

So, the inverse function is:

f-¹(x) = 2, x = 0

1/(x - 2), x ≠ 0 and x ≠ 2

Step-by-step explanation:

Answer:

A.

Step-by-step explanation:

Answer:

A

Step-by-step explanation:

hope it helps you

Answer:

5=x

Step-by-step explanation:

Q=2(2x-5)=3x+x-2x

SOLUTION:

4x-10=4x-2x

-10=2x-4x

-10= -2x

(-+-=+ so 10 and 2 are positive now)

10/2=x

5=x

Answer:

-12x⁸y⁵

Step-by-step explanation:

Exponent law:

         xᵃ * xᵇ = xᵃ ⁺ ᵇ

In exponent multiplication, if bases are same, add the exponents.

(4x⁵y³) (-3x³y²) = 4*(-3) * x⁵⁺³ * y³⁺²

                        = -12x⁸y⁵

Answer:

slope = 0

Step-by-step explanation:

calculate the slope m using the slope formula

m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]

with (x₁, y₁ ) = (2, 3 ) and (x₂, y₂ ) = (5, 3 )

m = [tex]\frac{3-3}{5-2}[/tex] = [tex]\frac{0}{3}[/tex] = 0

a line with a slope of zero is a horizontal line parallel to the x- axis

The statement means that an n-ary operation on set X takes n inputs and produces a single output, with the operation defined by the function f:[tex]X^n[/tex] -> X.

The statement "An n-ary operation on x is functions f:X^n -> X" can be understood as follows:

In mathematics, an n-ary operation refers to an operation that takes n inputs and produces a single output. In this case, the operation is defined on a set X, and the inputs are elements from X. The result of the operation is also an element from X.

The symbol "f" represents the function that performs the n-ary operation. The function f takes n arguments, which are elements from the set X. The notation [tex]X^n[/tex] indicates the Cartesian product of X with itself n times, meaning that the inputs are taken from X and repeated n times as ordered tuples. The arrow "->" signifies that the function f maps the n-tuple of inputs to a single element in X.

To put it simply, the statement is describing a mathematical operation that takes n elements from a set X and combines them in some way to produce a single element, where the combination is determined by the function f. This function f can be defined explicitly based on the specific operation being considered.

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Answer:

451$

Step-by-step explanation:

$1,204 + $600 = 1804$

Since there are 4 members on the team, we divide the total sum of costs by 4.

$1804/4 = 451$.

Answer:

The athletic department spent $1,204 on airline tickets and $600 on hotel costs for the 4 members of the tennis team. To find out how much was spent for each member of the tennis team, you can add the cost of airline tickets and hotel costs and then divide by the number of members.

So, the total cost for airline tickets and hotel costs is $1,204 + $600 = $1,804. Since there are 4 members on the tennis team, the cost per member is $1,804 ÷ 4 = $451.

Therefore, $451 was spent for each member of the tennis team.

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The expression to show how much it will cost if Kenny buys x number of hotdogs is x * $2.

To find the total cost if Kenny buys x number of hotdogs, we can multiply the number of hotdogs (x) by the cost per hotdog ($2). The expression to calculate the cost would be:

Total Cost = x * $2

In this expression, x represents the number of hotdogs Kenny wants to buy, and $2 represents the cost per hotdog.

For example, if Kenny wants to buy 5 hotdogs, we can substitute x with 5 in the expression:

Total Cost = 5 * $2

Simplifying the expression:

Total Cost = $10

So, if Kenny buys 5 hotdogs, it will cost him $10 in total.

Similarly, if Kenny buys any other number of hotdogs, we can substitute that value for x in the expression to calculate the corresponding total cost.

In summary, the expression to show how much it will cost if Kenny buys x number of hotdogs is x * $2.

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Answer:

Step-by-step explanation:

Given function is

[tex]f(x) = \dfrac{\left(x^2-9\right)}{\left(x^2-x-6\right)}[/tex]

Part 1

Graph attached

y-intercepts can be found by finding f(0) ie the value of f(x) at x = 0

[tex]f(0) = \dfrac{\left(0^2-9\right)}{\left(0^2-0-6\right)} = \dfrac{-9}{-6} = \dfrac{3}{2}[/tex]

Roots of a function can be found by setting f(x) = 0 and solving for x

Setting f(x) = 0

==> [tex]\dfrac{x^2-9}{x^2-x-6} = 0\\\\[/tex]

We can factor the numerator  as follows:
x² - 9 = (x + 3) (x -3)     since (a + b)(a-b) = a² - b²

Denominator can be factored as follows
x² - x - 6 = (x-3)(x+2)

So
[tex]f(x) = \dfrac{(x + 3)(x-3)}{(x-3)(x+2)}\\\\[/tex]

The (x-3) term cancels leaving
[tex]f(x) = \dfrac{x+3}{x+2}[/tex]

Setting this equal to 0 gives
[tex]\dfrac{x+3}{x+2} = 0[/tex]

This is 0 when x + 3 = 0 or x = -3

So there is only one root and that is x = -3

Asymptotes

The vertical asymptote occurs when at a value of x when the denominator becomes 0

The given function has been factored as
[tex]f(x) = \dfrac{x + 3}{x + 2}[/tex]

The denominator becomes 0 at x = -2
Vertical asymptote is x = - 2

To find the horizontal asymptote use the fact that when the degrees of the numerator and denominator are equal, the horizontal asymptote is given by
[tex]y=\dfrac{\mathrm{numerator's\:leading\:coefficient}}{\mathrm{denominator's\:leading\:coefficient}}[/tex]

The degree of the numerator x + 3 is 1 and the degree of the denominator x + 2 is also 1

So the horizontal asymptote is y = 1/1 = 1

y = 1 is the horizontal asymptote

End behavior is the behavior of the function as x → ±∞  

This is determined by examining the leading term of the function and determining what its behavior is as x → ±∞

In the function
[tex]f(x) = \dfrac{x + 3}{x + 2}[/tex]
which is the factored form of the originally given function
the domain of x = all real numbers with the exception of -2 since at x = -2, the function is undefined

The end behavior can be determined by finding the limit of f(x) as x tends to infinity

[tex]\lim _{x\to \infty \:}\left(\dfrac{x+3}{x+2}\right)\\\\\dfrac{x+3}{x+2} \text{ can be transformed by dividing by x both numerator and denomiator :}\\\\\\=\dfrac{\dfrac{x}{x}+\dfrac{3}{x}}{\dfrac{x}{x}+\dfrac{2}{x}}\\\\\\=\dfrac{1+\frac{3}{x}}{1+\dfrac{2}{x}}[/tex]

[tex]\lim _{x\to \infty \:}\left(\dfrac{x+3}{x+2}\right) \\\\\\\\=\lim _{x\to \infty \:}\left(\dfrac{1+\dfrac{3}{x}}{1+\dfrac{2}{x}}\right)\\\\\\\\=\dfrac{\lim _{x\to \infty \:}\left(1+\dfrac{3}{x}\right)}{\lim _{x\to \infty \:}\left(1+\dfrac{2}{x}\right)}[/tex]

[tex]\lim _{x\to \infty \:}\left(1+\dfrac{3}{x}\right) = 1\\\\\lim _{x\to \infty \:}\left(1+\dfrac{2}{x}\right) = 1[/tex]

[tex]\lim _{x\to \:-\infty \:}\left(\dfrac{x+3}{x+2}\right) = 1[/tex]

End behavior
[tex]\mathrm{as}\:x\to \:+\infty \:,\:y\to \:1,\:\:\mathrm{and\:as}\:x\to \:-\infty \:,\:y\to \:1[/tex]

Table:
x                y

-4            1/2

- 3            0

-1               2

0               3/2

1                4/3

2                5/4

3               6/5

Note that the function is undefined at x = -2

Using the first two points we can get an exponential model:

[tex]y = (225/23)*(23/15)^x[/tex]

The general exponential equation can be written as:

[tex]y = A*(b)^x[/tex]

We can replace any two values of the table to get the system of eqautions:

[tex]15 = A*(b)\\\\23 = A*(b)^2[/tex]

Taking the quotient between the second and the first equation we get:

[tex]23/15 = (A*b^2)/(A*b)\\\\23/15 = b[/tex]

Replacing that on the first equation we will get:

[tex]15 =A*(23/15)\\15*(15/23) = A\\225/23 = A[/tex]

Then an exponential model is:

[tex]y = (225/23)*(23/15)^x[/tex]

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Answer:

0.24ft

7m

Step-by-step explanation:

Answer:

We can simplify the expression log₂8³x using the following logarithmic rule:

logₐb^c = c * logₐb

Using this rule, we can rewrite the expression as:

log₂8³x = 3log₂8 + log₂x

We can further simplify by recognizing that 8 can be expressed as a power of 2:

8 = 2³

Substituting this in, we get:

3log₂8 + log₂x = 3log₂(2³) + log₂x = 3 * 3 + log₂x = 9 + log₂x

Therefore, the simplified expression is:

log₂8³x = 9 + log₂x

Step-by-step explanation:

Answer:

0.01167 m/s^2

Step-by-step explanation:

Acceleration is chage in velocity with respect to time. Another way to think of acceleration is in the units: m/s^2, or as I like to say, (m/s) /s.

The strategy to solve this kind of problem is to find the change in velocity (which will be in units of m/s)... and then divide that by the time passed.

In this case:

Change in Velocity = (13 m/s) - (6 m/s) = 6 m/s

Time passed = 10 min

Converting units of time: 10 min * (60 s / 1 m) = 600 sec

Acceleration = Change in Velocity / Time Passed

Acceleration = (6 m/s) / (600 s)

Acceleration = 0.01167 m/s^2

James drank [tex]$27 \frac{1}{2}$[/tex] cups of milk during the week

Therefore, James drank [tex]$27 \frac{1}{2}$[/tex] cups of milk during the week.

NOTE: The given question has wrong information. The correct question is:

"At the beginning of the week, James has 8 quarts of milk. At the end of the week, there are [tex]$4 \frac{1}{2}$[/tex] cups of milk left. How many cups of milk did James drink during the week?"

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The kite string is 331.77 feet long.

Trigonometry deals with the relationship between the ratios of the sides of a right-angled triangle with its angles.

Check the attached image for labeling. Where l represents the length of kite string.

h = 225 - 3 = 222 ft

Using trig. ratio:

sin 42° = 222/l

l = 222 / sin 42°

l = 331.77 feet

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No, the inference is not valid because it excluded those who were not online and so did not see the poll.

To improve the poll, the news show could have conducted it in a more representative manner by surveying a larger sample of people from a variety of backgrounds

The poll's results may have been influenced by various factors, including but not limited to the order in which the comedians were presented, the demographics of the respondents, and the fact that it was conducted online, possibly excluding those without internet access. Overall, the poll's representative accuracy is questionable.

To enhance the accuracy of the survey, the broadcast could have implemented a more inclusive method by polling a wider range of individuals from diverse backgrounds. An alternate method for conducting the poll would have been by personal or telephonic means, enabling individuals without internet connectivity to participate.

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The solution set of given inequalities are represented by Part A.

The given inequalities are

⇒ y ≥ x + 1 and y + x > -1

Hence, The related equations of both inequalities are

y = x + 1

Put x=0, to find the y-intercept and put y=0, to find x intercept.

y = 0 + 1

y = 1

And, 0 = x + 1

x = - 1

Therefore, x-intercept of the equation is (-1,0) and y-intercept is (0,1).

Similarly, for the second related equation

y + x = - 1

y + 0 = - 1

y = - 1

0 + x = - 1

x = - 1

Therefore x-intercept of the equation is (-1,0) and y-intercept is (0,-1).

Now, join the x and y-intercepts of both lines to draw the line.

Now check the given inequalities by (0,0).

0 ≥ 0 + 1

0 ≥ 1

It is a false statement, therefore the shaded region is in the opposite side of origin.

0 + 0 ≥ - 1

0 ≥ - 1

It is a true statement, therefore the shaded region is about the origin.

Hence, From the below figure we can say that the solution set of given inequalities are represented by Part A.

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Answer:she can make three 5 ounce candles. she had 15 ounces left over , 15/5=3 .

Step-by-step explanation: 5 of 10% of 20 ounces is 10 ounces, and both the 5 five ounce candles and the twenty five one ounce candles are each 2.5 extra ounces left. (5 for both), making a totalm of 15 ounces left to put into new candles.

It is shown below that the partition of G into left cosets of H is different from its partition into right cosets when n ≥ 3.

Let us first find the left cosets of H in G:

H = {1, s₀}

s₁H = {s₁, s₁s₀}

s₂H = {s₂, s₂s₀}

s₃H = {s₃, s₃s₀, s₂s₁, s₂s₁s₀}...

Now, find the right cosets of H in G:

H = {1, s₀}

Hs₁ = {s₁, s₀s₁}

Hs₂ = {s₂, s₀s₂}

Hs₃ = {s₃, s₀s₃, s₁s₂, s₀s₁s₂}...

Observe that the left cosets of H in G and the right cosets of H in G are not the same when n≥3. This is because s₁s₀ is not in H, but s₀s₁ is in H.

Therefore, the left coset s₁H is not the same as the right coset Hs₁. Similarly, s₂s₁s₀ is not in H, but s₀s₁s₂ is in H, so the left coset s₃H is not the same as the right coset Hs₃.

Thus, it is shown that the partition of G into left cosets of H is different from its partition into right cosets when n≥3.

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Answer:

1. x = 13.75

2. 17 degrees

3. a = 4, b = 2√2

Step-by-step explanation:

1. cos(37) = 11/x

since cos(37) = 4/5

4/5 = 11/x

4x = 55

x = 13.75

2. tan(0) = opp/adj

tan(0) = 4/13 = 0.30769

=> (0) = 17.1 degrees or 17 degrees

3. since the triangle is right triangle with 2 angles of 45 degress

=> isosceles right triangle

then b = 2√2

a^2 = (2√2)^2 + (2√2)^2

a^2 = = 8 + 8 = 16

a = √16= 4

The length of gateleg of the table whose leg swings from one corner of the table to the middle of the far side of the table top is 53 inches long.

The shape forms a trapezoid

The length of the parallel side = is 56 in

The length of the base = is 45 in

To the length of the triangle formed  when the base of the gateleg is connected with the middle of the table

The length of the side when connected with the midpoint = 56/2 = 28

The length of gateleg = hypotenuse

Using Pythagorean theorem :

(hypotenuse)² = (base)² + (perpendicular)²

(hypotenuse)² = (45)² + (28)²

(hypotenuse)² = 2025 + 784

(hypotenuse)² = 2809

hypotenuse = 53

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The question is incomplete the complete question is :

Philippa has a gateleg table in her kitchen. The leg swings from one corner of the table to the middle of the far side of the tabletop, as shown in the sketch below.

Note: The figure is not drawn to scale.

How long is the gateleg of the table?

A.) 53 inches

B.) 49 inches

C.) 73 inches

D.) 72 inches

The percentage of the data values that are greater than 130 is 25%

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

The box plot

From the box plot, the five-number summary are

40, 70, 110, 130, and 150

In the above five-number summary, we have

Third quartile = 130

This means that the percentage of the data values that are greater than 130 is 25%

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The 5th term in the expansion of [tex](3x + 1)^7[/tex] would be [tex]945x^3[/tex].

To find the 5th term in the expansion of  [tex](3x + 1)^7[/tex], we use the binomial theorem formula:

[tex](a + b)^n = nC0 a^n b^0 + nC1 a^(n-1) b^1 + nC2 a^(n-2) b^2 + ... + nCn-1 a^1 b^(n-1) + nCn a^0 b^n[/tex]

where nCk represents the binomial coefficient, given by the formula:

nCk = n! / (k!(n-k)!)

In this case, a = 3x and b = 1, so we have:

[tex](3x + 1)^7 = 7C0 (3x)^7 1^0 + 7C1 (3x)^6 1^1 + 7C2 (3x)^5 1^2 + 7C3 (3x)^4 1^3 + 7C4 (3x)^3 1^4 + 7C5 (3x)^2 1^5 + 7C6 (3x)^1 1^6 + 7C7 (3x)^0 1^7[/tex]

To find the 5th term, we need to look at the term with k = 4, which is:

[tex]7C4 (3x)^3 1^4[/tex] = [tex]35 (3x)^3[/tex]

     = [tex]35 (3x)^3[/tex]

     = [tex]35 (27x^3)[/tex]

Therefore, the 5th term in the expansion of [tex](3x + 1)^7[/tex] is [tex]945x^3[/tex].

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The probability of event W occurring given that event R has occurred is 6/19. Among the given answer choices, the correct probability is 6/19.

To find P(W|R), we can use the conditional probability formula:

P(W|R) = P(W & R) / P(R)

Given that P(R) = 1/3 and P(R & W) = 2/19, we can substitute these values into the formula:

P(W|R) = (P(R & W)) / P(R) = (2/19) / (1/3)

To divide by a fraction, we can multiply by its reciprocal

P(W|R) = (2/19) * (3/1) = 6/19

Therefore, the probability of event W occurring given that event R has occurred is 6/19.

Among the given answer choices, the correct probability is 6/19.

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The percentage of refrigerators that have lives between 11 years and 18 years is approximately 83.01%.

The values of 11 years and 18 years using the given mean and standard deviation, and then find the area under the standard normal curve between those two standardized values.

First, we standardize the value of 11 years:

z1 = (11 - 14) / 2.5 = -1.2

Next, we standardize the value of 18 years:

z2 = (18 - 14) / 2.5 = 1.6

Now we need to find the area under the standard normal curve between these two standardized values.

We can use a standard normal table or calculator to find this area.

Using a standard normal table, we can find the area between z = -1.2 and z = 1.6 by finding the area to the left of z = 1.6 and subtracting the area to the left of z = -1.2:

Area = P(-1.2 < z < 1.6) = P(z < 1.6) - P(z < -1.2)

Looking up these values in the standard normal table, we find:

P(z < 1.6) = 0.9452

P(z < -1.2) = 0.1151

Substituting these values, we get:

Area = 0.9452 - 0.1151 = 0.8301

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The solution of the system of equations is (-2, 2).

Here we want to find the solution of the system of equations:

x + 2y = 2

y = -2x- 2

To solve this, we can replace the second equation into the first one, then we will get:

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

Now we can solve that for x:

x - 4x - 4 = 2

-3x = 2 + 4

-3x = 6

x = 6/-3 = -2

Thus, the value of y is:

y = -2*-2 - 2

y = 4 - 2 = 2

The solution of the system is (-2, 2)

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Complete question:

"Which is the solution of the system of equations?

x + 2y = 2

y = -2x- 2"

Nancy's bag was 0.2kg over the weight limit.

Let's denote the weight limit as x.

The total cost for the first airline can be expressed as:

Cost1 = 20x + 5

The total cost for the second airline is given by:

Cost2 = 25x + 4

As fees were same with both airlines, we will set up equation to find the weight difference:

20x + 5 = 25x + 4

20x-25x = 4-5

-5x = -1

x = -1/-5

x = 0.2kg

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A number line that shows the solutions to x+8 > 15 include the following: B. number line B.

In Mathematics and Geometry, an inequality simply refers to a mathematical relation that is typically used for comparing two (2) or more numerical data and variables in an algebraic equation based on any of the inequality symbols;

Based on the information provided above, we have the following equation (inequality);

x + 8 > 15

By subtracting 8 from both sides of the equation (inequality), we have;

x + 8 - 8 > 15 - 8

x > 7

This ultimately implies that, the inequality must be shaded to the right with an open dot because the inequality symbol is greater than (>).

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Missing information:

The question is incomplete and the complete question is shown in the attached picture.