If the sum of three consecutive numbers is 2022, what is the first numebr?

Substituting the value of \( r \), we get \( V \approx 1105.4 \) mm³. Finally, dividing the mass of the ball-bearing (11.9 g) by its volume (1105.4 mm³) and converting the units, we can determine the density in g/cm³. The density is approximately 0.0108 g/cm³.

To explain the process in more detail, we start by finding the radius of the ball-bearing using the circumference formula. The circumference is given as 43.4 mm, so dividing it by 2π gives us the radius of approximately 6.912 mm.

Next, we calculate the volume of the sphere using the formula \( V = \frac{4}{3}\pi r^3 \). Plugging in the radius value, we obtain the volume of the metal ball-bearing as approximately 1105.4 mm³.

To calculate the density, we divide the mass of the ball-bearing (11.9 g) by its volume (1105.4 mm³). However, to obtain the density in g/cm³, we need to convert the volume from mm³ to cm³ by dividing it by 1000. After performing the division and conversion, we find the density of the metal ball-bearing to be approximately 0.0108 g/cm³.

Density is a fundamental property of matter that describes how much mass is contained within a given volume. In this case, it allows us to understand the mass-to-volume ratio of the metal ball-bearing. By calculating the density, we can characterize the compactness or heaviness of the material.

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The density of the metal in the ball-bearing is approximately 10.981 g/cm^3.

To find the density of the metal in g/cm^3, we need to calculate the volume of the metal ball-bearing and divide it by its mass.

Given information:

Circumference of the ball-bearing = 43.4 mm

Weight of the ball-bearing = 11.9 g

To calculate the volume of the ball-bearing, we need to find its radius (r). We can use the formula for the circumference of a circle:

Circumference = 2πr

Substituting the given circumference

43.4 mm = 2πr

To find the radius, divide both sides by 2π:

r = 43.4 mm / (2π) ≈ 6.9134 mm

Next, let's convert the radius to centimeters:

r = 6.9134 mm / 10 ≈ 0.69134 cm

Now we can calculate the volume of the ball-bearing using the formula for the volume of a sphere:

V = (4/3)πr^3

Substituting the radius:

V = (4/3)π(0.69134 cm)^3

Calculating this expression:

V ≈ 1.083 cm^3

Finally, to find the density, we divide the mass by the volume:

Density = Mass / Volume

Density = 11.9 g / 1.083 cm^3

Calculating this expression:

Density ≈ 10.981 g/cm^3

Therefore, the density of the metal in the ball-bearing is approximately 10.981 g/cm^3.

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\(~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$11250\\ r=rate\to 3.45\%\to \frac{3.45}{100}\dotfill &0.0345\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{annually, thus once} \end{array}\dotfill &1\\ t=years\dotfill &5 \end{cases}\)

\(A=11250\left(1+\frac{0.0345}{1}\right)^{1\cdot 5} \implies A \approx 13329.23 ~\hfill \underset{interest~paid}{\stackrel{13329.23~~ - ~~11250}{\approx\text{\LARGE 2079.23}}}\)

Answer:

Use you calculator.

Step-by-step explanation:

There will be a way on your calculator to do tan^-1. It may be called arctan. They are the same thing.

Answer:

3(2x-5)+4(x+2) =33

Step-by-step explanation:

3(2x-5)+4(x+2)

3(2(4)-5)+4(4+2)

3(8-5)+4(6)

3(3)+24

9+24

33

Hope this helps♥

Answer:

12

Step-by-step explanation:

The average rate of change of f(x) in the closed interval [ a, b ] is

\(\frac{f(b)-f(a)}{b-a}\)

Here [ a, b ] = [ 3, 6 ] , then

f(b) = f(6) = 6² + 3(6) + 2 = 36 + 18 + 2 = 56

f(a) = f(3) = 3² + 3(3) + 2 = 9 + 9 + 2 = 20 , then

average rate of change = \(\frac{56-20}{6-3}\) = \(\frac{36}{3}\) = 12

Answer:

12

Step-by-step explanation:

Answer:

The value of x is 15.75

Explanation:

Given that triangles A and B are similar.

Side lengths of triangle A are: 4.5, 7.0, and 5.25

Side lengths of triangle B are: 13.5, 21.0, and x

Since the triangles are similar, the following ratios are correct:

13.5/4.5 = 21/7 = x/5.25

3 = x/5.25

Multiply both sides of the equation by 5.25

x = 15.75

The percentage change in temperature is 19.0%.

The percentage is a ratio that can be expressed as a fraction of 100.

Given that, the temperature rises from 21c to 29c.

The change in temperature is:

29c - 21c

= 4c

The percentage change in temperature from 21c is:

percentage change = (4c/21c)×100

                                 =4/21×100

                                 = 19.0%

Therefore, the percentage change in temperature is 19.0%.

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Answer:

Hope this will help you

Remember that

The volume of a sphere is equal to

we have that

the diameter is reduced to half its original size

D=D/2

so

the new radius is

r=r/2

substitute in the formula of volume

The probability that a single bulb will burn out within the next 10 minutes is approximately 0.6321. The expected amount of time until the first bulb goes out is 10 minutes. The probability density function (pdf) of the random variable T, representing the time when you leave the room after the last light bulb goes out, is given by \(g(t) = 20 * (1/10) * e^{(-(1/10)t)} * (1 - e^{(-(1/10)t))^(19)}\).

To find the probability that a single bulb will burn out within the next 10 minutes, we can use the exponential distribution. The exponential distribution with a mean of 10 minutes has a rate parameter λ = 1/10.

The probability density function (pdf) for an exponential distribution is given by \(f(x) = λ * e^{(-λx)}\)

In this case, we want to find the probability that a bulb burns out within the next 10 minutes, which corresponds to the cumulative distribution function (CDF) at x = 10. The CDF is given by \(F(x) = 1 - e^{(-λx)\)

So, substituting the values, we have:

\(F(10) = 1 - e^{(-(1/10)*10)\)

\(= 1 - e^{(-1)\)

= 1 - 0.3678794412

≈ 0.6321

Therefore, the probability that a single bulb will burn out within the next 10 minutes is approximately 0.6321.

The amount of time until the first bulb goes out follows an exponential distribution with a rate parameter of λ = 1/10 (since it has a mean of 10 minutes).

The probability density function (pdf) for the time until the first bulb goes out is given by\(f(t) = λ * e^{(-λt).\)

To find the expected amount of time until the first bulb goes out, we need to calculate the mean (or expected value) of this distribution.

The expected value of an exponential distribution with rate parameter λ is equal to 1/λ. In this case, the expected value is 1/(1/10) = 10 minutes.

Therefore, the expected amount of time until the first bulb goes out is 10 minutes.

To find the probability density function (pdf) of the random variable T, which represents the time when you leave the room (after the last light bulb goes out), we need to consider the distribution of the maximum of the exponential random variables.

Since there are 20 light bulbs in the room, and each follows an exponential distribution with a rate parameter λ = 1/10, the time until the last bulb goes out can be modeled as the maximum of 20 exponential random variables.

The pdf of the maximum of independent exponential random variables with the same rate parameter λ is given by \(g(t) = n * λ * e^{(-λt)} * (1 - e^{(-λt))^(n-1)}\), where n is the number of random variables.

In this case, n = 20, and λ = 1/10. Thus, the pdf of T is \(g(t) = 20 * (1/10) * e^{(-(1/10)t)} * (1 - e^{(-(1/10)t))^(19)}\)

This expression represents the pdf of the random variable T, which denotes the time when you leave the room after the last light bulb goes out.

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Answer: y = 45x + 345

Step-by-step explanation:

Hope this helps!!! :)

An equation of a circle that contains the set of points is (x - 6)² + (y + 2)² = 650.

In Mathematics and Geometry, the standard form of the equation of a circle is modeled by this mathematical equation;

(x - h)² + (y - k)² = r²

Where:

By using the distance formula, we would determine the radius based on the center (6, -8) and one of the given points (11, 17);

Radius (r) = √[(x₂ - x₁)² + (y₂ - y₁)²]

Radius (r) = √[(11 - 6)² + (17 + 8)²]

Radius (r) = √[25 + 625]

Radius (r) = √650 units.

By substituting the center (6, -2) and radius of √650 units, we have:

(x - 6)² + (y - (-2))² = (√650)²

(x - 6)² + (y + 2)² = 650

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The width and length of the rectangle is 27 cm and 36 cm. respectively.

To solve the problem, we can use the Pythagorean theorem since the diagonal, width, and length of the rectangle form a right triangle. The Pythagorean theorem states that:
a² + b² = c²
where a and b are the legs of the right triangle and c is the hypotenuse (diagonal of the rectangle).

Let's denote the width of the rectangle as w, the length as l, and the diagonal as d. According to the problem, we have:
d = w + 18
l = w + 9

Substituting these equations into the Pythagorean theorem, we get:
(w + 9)² + w² = (w + 18)²

Expanding and simplifying the equation, we get:
w² - 18w - 243 = 0

We can solve for w by factoring the expression:
(w - 27)(w + 9) = 0

w = 27 or w = -9

The positive solution for w is 27. We can use this value to find the length of the rectangle:
l = w + 9
l = 27 + 9
l = 36

Therefore, the width of the rectangle is 27 cm, and the length of the rectangle is 36 cm.

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Answer:

After 15 days they'll both have $10

Explanation:

100 (Xavier's total)- 6x (6 dollars every day) = 160 (Zach's total amount) -10x (10 dollars every day)

\(100-6x=160-10x\\\)

+10x on both sides

\(100+4x=160\)

-100 on both sides

\(4x=60\)

/4 on both sides

\(x=15\)

Answer: 4.2 miles

Leila running rate = 6 miles/ 50 minutes = 3/25 mi per minute

Rate = distance/ time

3/25 = distance / 35

3/25 * 35 = distance

4.2 = distance

so she can run 4.2 miles

The following differential equations with a solution
y" + y = 0 corresponds to solution A: y = cos(2)

y" + y' + 28y = 0 corresponds to solution B: y = e^(-4x)

y" + 11y' + 28y = 0 corresponds to solution C: y = e^(7x)

2x^2y" + 3xy' = y corresponds to solution D: y = x^(1/2)

1. y" + y = 0:

This is a second-order homogeneous differential equation with constant coefficients. The characteristic equation is r^2 + 1 = 0, which has complex roots r = ±i. The general solution is therefore a linear combination of sine and cosine functions:

y = c1 cos(x) + c2 sin(x)

Using the initial condition y(0) = 1, we can solve for the constants to get:

y = cos(x)

2. y" + y' + 28y = 0:

This is a second-order homogeneous differential equation with constant coefficients. The characteristic equation is r^2 + r + 28 = 0, which has complex roots given by the quadratic formula:

r = (-1 ± sqrt(1 - 4*28)) / 2 = (-1 ± 7i) / 2

The general solution is therefore a linear combination of exponential and sine/cosine functions:

y = e^(-x/2) (c1 cos(7x/2) + c2 sin(7x/2))

Using the initial condition y(0) = 1, we can solve for the constants to get:

y = e^(-4x)

3. y" + 11y' + 28y = 0:

The characteristic equation is r^2 + 11r + 28 = 0, which can be factored as (r + 4)(r + 7) = 0. The roots are r = -4 and r = -7. Therefore, the general solution is a linear combination of exponential functions:

y = c1 e^(-4x) + c2 e^(-7x)

Using the initial condition y(0) = 1, we can solve for the constants to get:

y = e^(7x)

4. 2x^2y" + 3xy' = y:

Dividing both sides by x^2 and letting z = y/x^(1/2). Then, we get:

z' + (1/4x)z = 0

This is a first-order homogeneous differential equation with an integrating factor of e^(1/4 ln x) = x^(1/4). Multiplying both sides by the integrating factor, we get:

x^(1/4) z' + (1/4)x^(-3/4)z = 0

The left-hand side is the derivative of (x^(1/4) z), so we can integrate both sides to get:

x^(1/4) z = c1

Solving for z, we get:

z =c1/x^(1/4)

Substituting back for y, we get:

y = x^(1/2) z = c1 x^(1/4)

Using the initial condition y(1) = 1, we can solve for the constant to get:

y = x^(1/2)

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(a) The solution to the Algebraic Riccati Equation is  P = [q/v², 0; 0, 0].

(b) The optimal control input u is −vx1.

(c) The closed-loop poled of the resulting feedback system is stable.

(a) The solution to the Algebraic Riccati Equation is given by the equation P = Q − ATPA + ATPB(R + BTPB)−1BTAP, where P is the solution matrix, Q is the terminal cost matrix, A is the state matrix, B is the control matrix, and R is the control cost matrix. Using the given values, we have P = [q/v^2, 0; 0, 0], Q = 0, A = [0, 1; 0, 0], B = [0; 1], and R = 1. Plugging these into the equation, we get P = [q/v², 0; 0, 0].

(b) The optimal control input u is given by u = −(R + BTPB)−1BTAPx. Plugging in the given values and the solution for P from part (a), we get u = −vx1.

(c) The closed-loop poles of the resulting feedback system are given by the eigenvalues of the matrix A − BK, where K is the feedback gain matrix. Plugging in the given values and the solution for u from part (b), we get K = [0, v/q]. The eigenvalues of A − BK are λ1 = 0 and λ2 = −v/q, indicating that the system is stable.

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Answer:

Transitive Property of Congruence

Explanation:

Here, we want to get the congruence law used

From what we have, given 3 set of angles with 1 one of the angles equal to the two others, it mean that the other two are equal to one another

Thus, we can see that there is a linkage between the values as a result of one of the angles being equal to the other two

The name of the congruence property that represents this is the transitive property of congruence

We can use it to infer or get the value of a third angle or use to define the equality of two angles given a third one in which the first two are equal to

you do 479 x 2 = 958 there is your answer your welcome

the size of the union of the three sets is: |A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C| = 3  × 24 million - 3 × 1.4 million + 1.2 million ≈ 69 million

A combination is a way of selecting a subset of objects from a larger set without regard to their order. The formula for the number of combinations of n objects taken r at a time is:
C(n, r) = n! / (r! ×  (n - r)!)
where n! means the factorial of n, which is the product of all positive integers up to n. For example, 5! = 5 ×  4 ×  3 ×  2 ×  1 = 120.
To apply this formula to our problem, we first need to count the total number of coins in the pile. Since there are at least 16 of each type, the minimum total is:
16 + 16 + 16 + 16 = 64
But there could be more coins of each type, so the total could be larger than 64. However, we don't need to know the exact number, only that it is large enough to allow us to choose 16 coins from it.
Using the formula for combinations, we can calculate the number of different collections of 16 coins that can be chosen from the pile:
C(64, 16) = 64! / (16! × (64 - 16)!) ≈ 1.1 billion
That's a very large number! It means there are over a billion ways to choose 16 coins from a pile that contains at least 16 of each type.
To answer the second part of the question, we need to count the number of collections that contain at least 3 coins of each type. One way to do this is to use the inclusion-exclusion principle, which says that the number of elements in the union of two or more sets is equal to the sum of their individual sizes minus the sizes of their intersections, plus the sizes of the intersections of all possible pairs, minus the size of the intersection of all three sets, and so on.
In this case, we can consider three sets:
- A: collections with at least 3 pennies
- B: collections with at least 3 nickels
- C: collections with at least 3 dimes
- D: collections with at least 3 quarters
The size of each set can be calculated using combinations:
|A| = C(48, 13) ≈ 24 million
|B| = C(48, 13) ≈ 24 million
|C| = C(48, 13) ≈ 24 million
|D| = C(48, 13) ≈ 24 million
Note that we have to choose 3 coins of each type first, leaving 4 coins to be chosen from the remaining 48 coins.
The size of the intersection of any two sets can be calculated similarly:
|A ∩ B| = C(43, 10) ≈ 1.4 million
|A ∩ C| = C(43, 10) ≈ 1.4 million
|A ∩ D| = C(43, 10) ≈ 1.4 million
|B ∩ C| = C(43, 10) ≈ 1.4 million
|B ∩ D| = C(43, 10) ≈ 1.4 million
|C ∩ D| = C(43, 10) ≈ 1.4 million
Note that we have to choose 3 coins of each type first, leaving 1 coin to be chosen from the remaining 43 coins.
The size of the intersection of all three sets can also be calculated:
|A ∩ B ∩ C| = C(38, 7) ≈ 1.2 million
Note that we have to choose 3 coins of each type first, leaving 1 coin to be chosen from the remaining 38 coins.

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Answer:

Associative property

Step-by-step explanation:

you can group factors in any order and the answer will be the same

The position of the ball at t = 2 seconds is -0.8 meters.

The position of a ball rolling in a straight line is given by the equation x = 2.0 - 3.6t + 1.1t^2, where x is the position in meters and t is the time in seconds. To find the position of the ball at a specific time, we can plug in the value of t into the equation and solve for x.

For example, if we want to find the position of the ball at t = 2 seconds, we can plug in 2 for t and solve for x:

x = 2.0 - 3.6(2) + 1.1(2)^2
x = 2.0 - 7.2 + 4.4
x = -0.8

Therefore, the position of the ball at t = 2 seconds is -0.8 meters.

Similarly, we can plug in any value of t into the equation to find the position of the ball at that specific time.

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Answer:

A, an = 2n. B. an = 2(n-

Step-by-step explanation:

can you brainlist me

A mean is an arithmetic average of a set of observations.

The correct option is (A).

A mean is an arithmetic average of a set of observations.

Mean = (Sum of observations)/Number of observations

As, the median is the middle value.

Thus, the median will not be changed.

As, the mean is the arithmetic average.

So, the $100 gift will increase the mean value of the prizes.

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Answer:

A

Step-by-step explanation:

Step-by-step explanation:

slope = (y2-y1) / (x2-x1)

equation of the line y2 - y1 = m(x2-x1),

y2 = m(x2-x1) +y1

y2-y1 = -2(x2- (-3))

y2-y1 = -2x -6

y2 - 1 = -2x -6

y2 = -2x -5