A certain pipe can fill up a tank 2 hours faster than another pipe. It takes 3 and hours for both pipes to fill up the same tank. In how many hours would the first pipe fill up the tank?​

Answer:

2nd option

Step-by-step explanation:

Using \(a_{n}\) = 5\(a_{n-1}\) and a₁ = 2 then

a₂ = 5 × a₁ = 5 × 2 = 10

a₃ = 5 × a₂ = 5 × 10 = 50

the first 3 terms are 2, 10, 50

The first three terms of the given sequence are 2, 10, 50

A sequence in which every term is obtained by multiplying or dividing a definite number with the preceding number is known as a geometric sequence.

Given that, a sequence can be generated by using aₙ₋₁ = 5 and a₁ = 2,

We need to find the first three terms of the sequence,

Therefore, aₙ₋₁ = 5 and a₁ = 2 then

a₂ = 5 × a₁ = 5 × 2 = 10

a₃ = 5 × a₂ = 5 × 10 = 50

Therefore,

The first 3 terms are 2, 10, 50

Hence, the first three terms of the given sequence are 2, 10, 50

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Answer:

Step-by-step explanation:

3 x 4= 12 -5 = 7 + 4 = 11

a.  The function for the total expenditures is F(x) = 4.11x² + 87.23x + 1386.52

b.  In 2017, total expenditures will be 3669.57 billion dollars.

a. Since the rate of increase of total expenditures is given as f(x) = 8.22x + 87.23, the function that gives the total expenditures x years after 2000 can be found by integrating the rate of increase:

F(x) = ∫ f(x) dx = 4.11x² + 87.23x + C

Since the total expenditures were $1590.5$ billion in 2002, we can use this information to find the constant $C$:

F(2) = 4.11(2)² + 87.23(2) + C = 1590.5

Solving for C, we get:

C = 1386.52

Therefore, the function that gives the total expenditures x years after 2000 is:

F(x) = 4.11x² + 87.23x + 1386.52 (in billions of dollars)

b. To find the total expenditures in 2017, we need to substitute x = 17 in the function F(x):

F(17) = 4.11(17)² + 87.23(17) + 1386.52≈ 3669.57

Therefore, the total expenditures in 2017 will be approximately 3669.57 billion dollars.

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Answer:

Mean: 49.7

Median: 49.5

Mode: 24, 65

Range: 62

Step-by-step explanation:

Answer:

Step-by-step explanation:

Question 1: the defined rules dictates that series 1 is multiples of 3 and series 2 is multiples of 9, making the ratio always 3.

Question 2: You must pay attention to the values of the coins based on how many going from 0 to 3 so the nickels are 0, 5, 10, 15 and the values of dimes is 0, 10, 20, 10. Looking only at the values one could easily jump to answer A because the values initially double for each, but breaks down at 3 coins.

The slope of the line perpendicular to the given line is equal to -4/1.

The equation is in the form y=mx+b

so, y=(1/4)x -10

First, find the slope of the line which is the value being multiplied by x. Therefore m=1/4

To find the slope perpendicular to the line you must find the negative reciprocal of 1/4. (Which just means change the sign then flip the fraction)

Multiplying 1/4 by -1 which is -1/4

Then flip the fraction and keep the sign with the numerator to find the reciprocal. So now it’s -4/1

Thus, the slope of the line perpendicular to the given line is equal to -4/1.

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The equation is in the form y=mx+b

so, y=(1/4)x -10

First, find the slope of the line which is the value being multiplied by x. Therefore m=1/4

Multiplying 1/4 by -1 which is -1/4

Then flip the fraction and keep the sign with the numerator to find the reciprocal. So now it’s -4/1

Answer:

b

Step-by-step explanation:

It will take \(\frac{20}{9}\) hours as per linear equation to complete the wall.

A linear equation is an equation where the variable of the equation has the highest power of degree one.

Given, the 1st bricklayer build a complete wall of certain size in 5 hours.

Therefore, that bricklayer build \(\frac{1}{5}\)th of the complete wall in 1 hour.

Similarly, the 2nd bricklayer build a complete wall of certain size in 4 hours.

Therefore, that bricklayer build \(\frac{1}{4}\)th of the complete wall in 1 hour.

If they work together, in 1 hour they build a total of (\(\frac{1}{5}\)+\(\frac{1}{4}\))th = \(\frac{9}{20}\) of the complete wall.

Hence, they will complete the wall in (1÷\(\frac{9}{20}\)) hours = \(\frac{20}{9}\) hours.

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(a) The real discrete Fourier transform (DFT) is calculated for the given data set to analyze the helicopter's acoustic signature.

(b) To obtain the ck values and illustrate the relative size of each term in the Fourier series, we calculate the magnitude of each coefficient and plot their amplitudes on a bar graph against the corresponding frequency component, k.

To analyze the helicopter's acoustic signature, the real DFT is computed for the provided data set. The DFT transforms the time-domain measurements of acoustic pressure into the frequency domain, revealing the different frequencies present and their corresponding amplitudes. This analysis helps in understanding the spectral characteristics of the helicopter's acoustic signature and identifying prominent frequency components.

Using the Fourier series representation, the amplitudes (ck's) of the different frequency components in the Fourier series are determined. These amplitudes represent the relative sizes of each term in the series, indicating the contribution of each frequency component to the overall acoustic signature. By plotting the amplitudes on a bar graph, the relative strengths of different frequency components become visually apparent, enabling a clear comparison of their importance in characterizing the helicopter's acoustic signature.

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Answer:

x = 5, -5, 3, -3

Step-by-step explanation:

First, substitute u = x^2 into the equation, and we have

u^2 - 34u + 225 = 0

u = x^2

Factor u^2 −34u +225 using the AC method, and we get

( u - 25) ( u - 9) = 0

We know If any individual factor on the left side of the equation is equal to

0, the entire expression will be equal to 0, so

u - 25 = 0

u - 9 = 0

Next, we set u - 25 equal to 0  and solve for u, and we get

u = 25

Then set u - 9 equal to 0 and solve for u, and we get

u = 9

The final solution is all the values that make (u−25)(u−9) = 0  true.

u =25,9

Substitute the real value of u=x^2 back into the solved equation.

x^2 = 25

(x^2)^1 = 9

x^2 = 25

x = 5, -5

Now we solve the second equation

(x^2)^1 = 9

x = 3, -3

So, the answers are: 5, -5, 3, -3

The correct answer is k+3 ke-2.

To determine which formula is not equivalent to the other two, we can compare each formula step-by-step. Let's analyze each formula:

1. (-1k-3 k +3 k+ 4 ks4
2. k+3 ke-2
3. k-3 ks4

Starting with formula 1, we can see that it consists of three terms: -1k-3, k+3, and k+4ks4. Each term is separated by a space.

Moving on to formula 2, we have k+3ke-2. This formula also consists of three terms: k+3, k, and e-2. In this case, we have a variable (k) combined with two different exponents (3 and -2).

Finally, formula 3 is k-3ks4. It consists of two terms: k-3 and ks4. Again, each term is separated by a space.

Comparing the three formulas, we can see that formula 1 has three terms, while formulas 2 and 3 have two terms each. Additionally, formula 1 includes the term k+4ks4, which is not present in formulas 2 and 3.

Therefore, the formula that is not equivalent to the other two is k+3 ke-2.

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Answer:

6.5 inches

Step-by-step explanation:

1. The probability of selecting exactly one tank with high-viscosity material is 0.

2. The probability of selecting at least one tank with high-viscosity material is 1.

3. The probability of selecting exactly one tank with high-viscosity material and exactly one tank with high impurities is 0.25.

1. The probability of selecting exactly one tank with high-viscosity material is calculated by the binomial distribution formula, P(X = n) = (nCx)p^x(1-p)^n-x, where n is the number of trials, x is the number of successes, and p is the probability of success. In this case, n = 4, x = 1, and p = 24/24 = 1. Therefore, P(X = 1) = (4C1)1^1(1-1)^4-1 = 0.

2. The probability of selecting at least one tank with high-viscosity material is calculated by the complement rule, P(X > 0) = 1 - P(X = 0). In this case, P(X > 0) = 1 - (4C0)1^0(1-1)^4-0 = 1.

3. The probability of selecting exactly one tank with high-viscosity material and exactly one tank with high impurities is calculated by the binomial distribution formula, P(X = n) = (nCx)p^x(1-p)^n-x, where n is the number of trials, x is the number of successes, and p is the probability of success. In this case, n = 8, x = 2, and p = 24/24 = 1. Therefore, P(X = 2) = (8C2)1^2(1-1)^8-2 = 0.25.

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(a) To find the values of z for which the matrix does not have an inverse, we can set up the determinant of the matrix and solve for z when the determinant is equal to zero.

The given matrix is:

|x3  x|

|9   1|

The determinant of a 2x2 matrix can be found using the formula ad - bc. Applying this formula to the given matrix, we have:

Det = (x3)(1) - (9)(x) = x3 - 9x

For the matrix to have an inverse, the determinant must be non-zero. Therefore, we solve the equation x3 - 9x = 0:

x(x2 - 9) = 0

This equation has two solutions: x = 0 and x2 - 9 = 0. Solving x2 - 9 = 0, we find x = ±3.

So, the values of x for which the matrix has no inverse are x = 0 and x = ±3.

(b) (i) To find the size of the acute angle between the vectors (3,0) and (5,5), we can use the dot product formula:

u · v = |u| |v| cos θ

where u and v are the given vectors, |u| and |v| are their magnitudes, and θ is the angle between them.

Calculating the dot product:

(3,0) · (5,5) = 3(5) + 0(5) = 15

The magnitudes of the vectors are:

|u| = sqrt(3^2 + 0^2) = 3

|v| = sqrt(5^2 + 5^2) = 5 sqrt(2)

Substituting these values into the dot product formula:

15 = 3(5 sqrt(2)) cos θ

Simplifying:

cos θ = 15 / (3(5 sqrt(2))) = 1 / (sqrt(2))

To find the acute angle θ, we take the inverse cosine of 1 / (sqrt(2)):

θ = arccos(1 / (sqrt(2)))

(ii) If the vector (-k, 3) is orthogonal to (5,5), it means their dot product is zero:

(-k, 3) · (5,5) = (-k)(5) + 3(5) = -5k + 15 = 0

Solving for k:

-5k = -15

k = 3

So, the value of k is 3.

(c) Let J be the linear transformation from R2 to R2 that reflects points in the horizontal axis and then scales them by a factor of 2. The matrix of J can be found by multiplying the reflection matrix and the scaling matrix.

The reflection matrix in the horizontal axis is:

|1  0|

|0 -1|

The scaling matrix by a factor of 2 is:

|2  0|

|0  2|

Multiplying these two matrices:

J = |1  0| * |2  0| = |2  0|

   |0 -1|   |0  2|   |0 -2|

So, the matrix of J is:

|2  0|

|0 -2|

Therefore, y = 2 and z = -2.

(d) The volume of a parallelepiped can be found by taking the dot product of two adjacent sides and then taking the absolute value of the result.

The adjacent sides of the parallelepiped P are (0,3,0)

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Answer:

\(\textsf{Slope-intercept form}: \quad y=-\dfrac{4}{3}x+\dfrac{5}{3}\)

\(\textsf{Standard form}: \quad 4x+3y=5\)

Step-by-step explanation:

Given linear equation:

\(y-11=-\dfrac{4}{3}(x+7)\)

The given linear equation is point-slope form.

\(\boxed{\begin{minipage}{3.7cm}\underline{Slope-intercept form}\\\\$y=mx+b$\\\\where $m$ is the slope\\ and $b$ is the $y$-intercept.\\\end{minipage}}\)

To write the equation in slope-intercept form, isolate y:

\(\implies y-11=-\dfrac{4}{3}(x+7)\)

\(\implies y-11=-\dfrac{4}{3}x-\dfrac{28}{3}\)

\(\implies y-11+11=-\dfrac{4}{3}x-\dfrac{28}{3}+11\)

\(\implies y=-\dfrac{4}{3}x+\dfrac{5}{3}\)

\(\boxed{\begin{minipage}{5.4 cm}\underline{Standard form}\\\\$Ax+By=C$\\\\where $A, B$ and $C$ are constants and $A$ must be positive.\\\end{minipage}}\)

To write the equation in standard form, eliminate the fraction, bring the terms in x and y to the left of the equation, and the constants to the right:

\(\implies y-11=-\dfrac{4}{3}(x+7)\)

\(\implies (y-11) \cdot 3=-\dfrac{4}{3}(x+7) \cdot 3\)

\(\implies 3y-33=-4(x+7)\)

\(\implies 3y-33=-4x-28\)

\(\implies 3y-33+4x=-4x-28+4x\)

\(\implies 3y-33+4x=-28\)

\(\implies 3y-33+4x+33=-28+33\)

\(\implies 4x+3y=5\)

The Hall family utilizes sprinklers for 15 hours compared to the 35 hours used by the Nguyen family.

The Hall family utilizes sprinklers for 15 hours compared to the 35 hours used by the Nguyen family.

Let h and n represent the amount of hours each household uses its sprinkler, the Halls and the Nguyens, respectively.

They use for a total of 50 hours, thus either h + n = 50 or h = 50 - n.

Additionally, their total water output is 1050 L, thus we may use the equation below.

35h + 15n = 1050

Replace h with h = 50 - n to get

35(50 - n) + 15n = 1050

1750 - 35n + 15n = 1050

35n - 15n = 1750 - 1050

20n = 700

34 hours, n

15 hours equal 50 h, 50 n, and 35 n.

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The volume of a cone with a diameter of 8 cm and a height of 6 cm is

100.48 cm³.

A cone is a three-dimensional geometric shape with a smooth and curving surface and a flat base, with an increase in the height radius of a cone decreasing to a certain point.

The volume of a cone is (1/3)πr²h.

The total surface area of a cone is πr(r + l).

The curved surface area is πrl.

Given, The height of the cone is 6 cm.

Also given the diameter of the base 2r = 8 cm hence r = 4 cm.

Therefore, The volume of this cone is,

= (1/3)π(4)²×6 cm³.

= 32π cm³.

= 100.48 cm³

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Um... is there a question?

Answer:

Step-by-step explanation:

- 15, -3, -1, 0 , 1, 5

Answer:

-15,-3,-1,0,1,5

Step-by-step explanation:

Answer:

Let the number be x,

x ≥ 4

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Thank You!!

Solution

By general rule w ehave this property:

Using this expression we can do the following:

Then the final simplfied form is:

(x-1)(x-4)

Then we can find the turning point like this:

And the corresponding y coordinate is:

Final answer:

The airspeed of the plane is approximately 377.8 mph and the speed of the wind is approximately 22.2 mph (rounded to one decimal place).

Let's denote the speed of the plane in still air as "p" and the speed of the wind as "w".

For the flight from Joppetown to Jawsburgh, the effective speed of the plane is reduced by the headwind. The time it takes for this leg of the journey is 4 hours and 30 minutes, which is equivalent to 4.5 hours.

Using the formula distance = speed * time, we can write the equation:

1600 = (p - w) * 4.5

For the return flight from Jawsburgh to Joppetown, the effective speed of the plane is increased by the tailwind. The time it takes for this leg of the journey is 4 hours.

Using the same formula, we can write the equation:

1600 = (p + w) * 4

We now have a system of two equations. Let's solve it to find the values of p and w.

From the first equation, we can express p - w as 1600 / 4.5. Simplifying, we get:

p - w = 355.56

From the second equation, we can express p + w as 1600 / 4. Simplifying, we get:

p + w = 400

Now, we can solve these two equations simultaneously.

Adding the two equations together, we eliminate w and get:

2p = 755.56

Dividing both sides by 2, we find:

p = 377.78

Substituting this value of p back into one of the equations, we can solve for w:

377.78 + w = 400

w = 400 - 377.78

w = 22.22

Therefore, the airspeed of the plane is approximately 377.8 mph and the speed of the wind is approximately 22.2 mph (rounded to one decimal place).

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=============================================

Explanation:

x = number of cups

y = total height

We have these ordered pairs

(x1,y1) = (2,16)

(x2,y2) = (4,20)

Let's find the slope of the line through those points.

\((x_1,y_1) = (2,16) \text{ and } (x_2,y_2) = (4,20)\\\\m = \text{slope} = \frac{\text{rise}}{\text{run}} = \frac{\text{change in y}}{\text{change in x}}\\\\m = \frac{\text{y}_{2} - \text{y}_{1}}{\text{x}_{2} - \text{x}_{1}}\\\\m = \frac{20 - 16}{4 - 2}\\\\m = \frac{4}{2}\\\\m = 2\\\\\)

A slope of 2, aka 2/1, means each time x goes up by 1, the y value goes up by 2.

It means each time we add a cup, the total height goes up by 2 cm.

This isn't the final answer, but it helps get us there.

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

We'll then use point-slope form to determine the equation.

\(y-y_1 = m(x - x_1)\\\\y-16 = 2(x - 2)\\\\y = 2(x - 2)+16\\\\y = 2x - 4+16\\\\y = 2x + 12\\\\\)

The last step is to plug x = 1 into that equation to determine the height of 1 cup.

So,

\(y = 2x + 12\\\\y = 2*1 + 12\\\\y = 2 + 12\\\\y = 14\\\\\)

We have x = 1 cup lead to a total height of y = 14 cm.

The final answer is 14 cm.

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

Notice that adding on 2 cm (the slope) gets us to 14+2 = 16 cm which is the height of two cups stacked together. A lot of the 2nd cup's height is part of the 1st cup's height. Only the 2 cm at the top is going to be added.

We could see that if we used x = 2.

\(y = 2x + 12\\\\y = 2*2 + 12\\\\y = 4 + 12\\\\y = 16\\\\\)

That confirms the left-most part of the diagram.

Now try x = 4.

\(y = 2x + 12\\\\y = 2*4 + 12\\\\y = 8 + 12\\\\y = 20\\\\\)

That works as well. It helps confirm we have the correct equation, and also confirms the final answer is correct. Another way to confirm is to either graph or use a table.

If you wanted to find the height of 8 cups, then,

\(y = 2x + 12\\\\y = 2*8 + 12\\\\y = 16 + 12\\\\y = 28\\\\\)

The proportion of days that are rainy is π (R) = 1/3.

The Markov chain for Exercise 31 is time-reversible if and only if it satisfies the condition of detailed balance.

Detailed balance implies that the product of the probabilities of each transition from one state to another in the forward and reverse directions is equal.

That is, for all states i, j,

Pijπi = Pjiπj

Here, the detailed balance equations for the given Markov Chain are:

π (S)P (S,C) = π (C)P (C,S)

π (S)P (S,R) = π (R)P (R,S)

π (C)P (C,S) = π (S)P (S,C)

π (C)P (C,R) = π (R)P (R,C)

π (R)P (R,S) = π (S)P (S,R)

π (R)P (R,C) = π (C)P (C,R)

By solving the above equations, we can find the probability distribution π as follows:

π (S) = π (C) = π (R)

= 1/3

In the long run, the proportion of days that are sunny is π (S) = 1/3.

And the proportion of days that are cloudy is also π (C) = 1/3.

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A process that behaves the same way as a procedure so that similar results are produced is called a simulation.

In this question,

Simulations are used for finding probabilities for compound events. The probability model of a random phenomenon consists of a sample space of possible outcomes, associated events, random variables, and a probability measure that specifies probabilities of events and determines distributions of random variables.

Procedure is a method of analyzing or representing statistical data; a procedure for calculating a statistic. The goal of statistical analysis is to identify trends.

Hence we can conclude that a process that behaves the same way as a procedure so that similar results are produced is called a simulation.

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The value of n when the sum is 143 is n = 12.

What is consecutive numbers?

The term "consecutive" refers to a sequence of numbers. Every pair of numbers has a difference of 1. The mean and median are equal in a set of consecutive numbers. If n is a number, n, n+1, and n+2 would be following it. Examples. 1, 2, 3, 4, 5.

Numbers that follow one another in order, from the smallest to the largest, are referred to as consecutive numbers. There is a consistent pattern to the difference between two consecutive numbers.

Given:

The sum of the first n consecutive odd integers can be found using

s = n^2 - 1, where n is greater than or equal to 1.

Let the sum is 143.

We have to find the value of n.

Plug s = 143 in s = n^2 - 1

⇒      143 = n^2 - 1

⇒ 143 + 1 = n^2

⇒      144 = n^2

⇒        12 = n

⇒         n = 12

Hence, the value of n when the sum is 143 is n = 12.

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Answer:

Step-by-step explanation:

-2.67

The value of the total area of the shaded region are,

⇒ 42 units²

We have to given that;

Sides of rectangle are,

AB = 9

BC = 6

Hence, The area of rectangle is,

⇒ 9 x 6

⇒ 54 units²

And, Area of triangle is,

A = 1/2 × 4 × 6

A = 12 units²

Thus, The value of the total area of the shaded region are,

⇒ 54 - 12

⇒ 42 units²

So, The value of the total area of the shaded region are,

⇒ 42 units²

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Answer:

2.74 x 10 square root of 3

Step-by-step explanation:

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