What are not changed after a rotation

Answer:

There are 20 cups of coffee in two pots of coffee and 36 donuts in 3 boxes of donuts

Step-by-step explanation:

The rule of three or is a way of solving problems of proportionality between three known values and an unknown value, establishing a relationship of proportionality between all of them. That is, what is intended with it is to find the fourth term of a proportion knowing the other three.

If the relationship between the magnitudes is direct, that is, when one magnitude increases, so does the other (or when one magnitude decreases, so does the other), the direct rule of three must be applied as follow:

a ⇒ b

c ⇒ x

So \(x=\frac{c*b}{a}\)

In this case, the rule of three can be applied as follows:

\(amount of cups of coffe=\frac{2coffe pots*10 cups of coffee}{1 coffe pot}\)

amount of cups of coffe= 20

\(amount of donuts=\frac{3 boxes*12 donuts}{1 box}\)

amount of donuts= 36

There are 20 cups of coffee in two pots of coffee and 36 donuts in 3 boxes of donuts

The answers are as follows:

This is a hypothesis test of a proportion, where the null hypothesis assumes that the proportion of people in the town suffering from depression or a depressive illness is greater than or equal to the national average of 9.5%, and the alternative hypothesis assumes that the proportion is less than 9.5%.

Since the sample size is 100 and the sample proportion is not too close to 0 or 1, a normal distribution can be used for the hypothesis test. The sample proportion is 0.07, and the standard deviation of the sample proportion is calculated using the formula for the standard deviation of a proportion.

The test statistic is the z-score, which measures how many standard deviations the sample proportion is from the null hypothesis proportion. The p-value is calculated as the probability of obtaining a z-score as extreme or more extreme than the observed z-score, assuming the null hypothesis is true.

The p-value is compared to the pre-conceived alpha level of 0.05, and if the p-value is less than alpha, the null hypothesis is rejected in favor of the alternative hypothesis. In this case, since the p-value is greater than alpha, we fail to reject the null hypothesis, and conclude that there is not enough evidence to suggest that the proportion of people in the town suffering from depression or a depressive illness is lower than the national average of 9.5%.

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The control valve in a hydraulic system is primarily used to control the flow rate of the fluid in a hydraulic circuit. This means it regulates the amount of fluid that passes through the system.

Additionally, the control valve can also be used to control the direction of fluid flow in the hydraulic circuit. By adjusting the position of the valve, the operator can determine the path that the fluid takes within the system.

However, the control valve is not directly responsible for controlling the fluid temperature or the pressure in the hydraulic circuit. These aspects are typically managed by other components such as heat exchangers or pressure relief valves.

To summarize, the control valve in a hydraulic system is mainly used to control the flow rate and direction of the fluid in the circuit. It does not directly control the fluid temperature or pressure.

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Using equation of 39.4 inches.,  the height of the candle after 26 hours will be 39.4 inches.

The graph of a linear function is a straight line. Both the independent and dependent variables of a linear function are one. X and Y are the independent and dependent variables, respectively.

Assume the candle will be 12 cm tall after 20 hours.

Use the formula: y - y1 = m(x - x1) where m = -.0.4; x = 18; y = 21.8 to find the point/slope:

y - 21.8 = -0.4 (x - 18)

The equation (y = length after x hours) is

y - 21.8 = -0.4 (x - 18)

y - 21.8 = -0.4 x + 7.2

y = -0.4x + 29

y = -0.4x + 29

the candle's height after 26 hours

After 15 hours, y = - 0.4(26) + 29

y = 10.4 + 29

y = 39.4 inches.

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The absolute minimum value of y = -5x^2 + 4x + 6 on the interval 1 ≤ x ≤ 2 is y = 1.

To find the absolute minimum, we need to evaluate the function at the critical points and endpoints within the given interval. First, let's find the critical points by taking the derivative of y with respect to x and setting it equal to zero: dy/dx = -10x + 4 = 0. Solving this equation, we get x = 2/5.

Next, we evaluate the function at the critical point and the endpoints of the interval. When x = 1, y = -5(1)^2 + 4(1) + 6 = 5. When x = 2/5, y = -5(2/5)^2 + 4(2/5) + 6 = 1. Finally, when x = 2, y = -5(2)^2 + 4(2) + 6 = -10.

Comparing these values, we find that y = 1 is the smallest value within the interval 1 ≤ x ≤ 2. Therefore, the absolute minimum value of the function y = -5x^2 + 4x + 6 on the given interval is y = 1.

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Answer:

2

Step-by-step explanation:

(-6)+(-12)=-18

(-18)+20=2

If it passes the horizontal lines test, it is a function.

A mathematical phrase, rule, or law that establishes the link between an independent variable and a dependent variable (the dependent variable). In mathematics, functions exist everywhere, and they are crucial for constructing physical links in the sciences.

Use the vertical line test to determine whether an expression is indeed a function; if so, the expression IS one.

In any case, if the graph passes the vertical line test first, indicating that it is a function, and IF then it also passes the horizontal line test, it indicates that it is not only a function but also a one-to-one function, meaning that there is a unique x-coordinate value for every unique y-coordinate value. However, if it passes the horizontal line test, it doesn't mean much functionally. What the heck is up with the horizontal line test anyway?

The inverse expression of a function is only true for one-to-one functions.

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Answer:

x = 7

Step-by-step explanation:

given 2 intersecting chords inside a circle then the product of the parts of one chord is equal to the product of the parts of the other chord , that is

16(5x - 11) = 38.4 × 10 = 384 ( divide both sides by 16 )

5x - 11 = 24 ( add 11 to both sides )

5x = 35 ( divide both sides by 5 )

x = 7

In a composite transformation as given you make first the transformation in the right and then the transformation in the left.

For the given point: (-7, 3)

1. Rotation 180°

2. Reflection over x-axis:

Answer:

Step-by-step explanation:

Therefore, the probability this student will receive 2 or more offers is 0.663 to 3 decimal places.

Since A and B are mutually exclusive, the probability of A occurring given that B has already occurred is 0. This means that P(A | B) = 0.

P(A ∩ B) = 0, since the events are mutually exclusive.

P(A | B) = 0, since the probability of A given B is 0 when A and B are mutually exclusive.

Since A and B are mutually exclusive, they cannot both occur at the same time. This means that the probability of them both happening at the same time is 0. As a result, P(A ∩ B) = 0.

Since A and B are mutually exclusive, the probability of A occurring given that B has already occurred is 0. This means that P(A | B) = 0.

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Answer:

x = 62 /9     y = 11 /9   z = − 5 /27

Step-by-step explanation:

The equation of the tangent line to the curve \(y = \frac {(x^2 - 1)}{ (x^2 + x + 1)}\) at the point (1, 0) is y = (2/3)x - 2/3.

To find the equation of the tangent line to the curve at the point (1, 0), we need to find the slope of the tangent line and then use the point-slope form of a linear equation.
Let's differentiate \(y = \frac {(x^2 - 1)}{ (x^2 + x + 1)}\) using the quotient rule:

\(y' = [(2x)(x^2 + x + 1) - (x^2 - 1)(2x + 1)] / (x^2 + x + 1)^2\)
Substituting x = 1 into the derivative expression:

\(y'(1) = [(2(1))(1^2 + 1 + 1) - (1^2 - 1)(2(1) + 1)] / (1^2 + 1 + 1)^2\)

        \(= [2(3) - (0)(3)] / (3)^2\)

        = 6/9

        = 2/3

Using the point-slope form y - y₁ = m(x - x₁), where (x₁, y₁) = (1, 0) and m = 2/3 we get,  
y - 0 = (2/3)(x - 1)

y = (2/3)x - 2/3

The point-slope form of a linear equation is given by y - y₁ = m(x - x₁) where (x₁, y₁) is a point on the line, and m is the slope of the line.

Therefore, the equation of the tangent line to the curve y = (x^2 - 1) / (x^2 + x + 1) at the point (1, 0) is y = (2/3)x - 2/3.

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The complete question is:

Find an equation of the tangent line to the given curve at the specified point, \(y = \frac {(x^2 - 1)}{ (x^2 + x + 1)}\) at (1,0).

The range of the function g(x) is given as follows:

[0, ∞).

From the graph of the function given in this problem, y assumes all real non-negative values, hence the interval notation representing the range of the function is given as follows:

[0, ∞).

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Answer:

a) Sinusoidal functions are y = a sin [b(x-h)] + k    (or)                                        

y = a cos [b(x-h)] + k

Where a is amplitude a= (max-min)/2=(16-2)/2=7

period p= 2π/b

            b=2π/30

            Horizontal transformation to 10 units right h=10

            k= (max+min)/2=(16+2)/2=9

h = 7 cos [π/15(t-10)]+ 9

b) t=10min=600 sec

substitue in the above equation

h=5.5m

Answer:

x>4

Step-by-step explanation:

x+3>19-3x

x+3x>19-3

4x>16

x>4

Answer:

x > 4

Step-by-step explanation:

Solution set:

Solution set as set builder notation:

Answer:

259.8

Step-by-step explanation:

The length of the curve is approximately 34.10 units.

To find the length of the curve described by the vector-valued function r(t) = 9t i + \(8t^{(3/2)}\) j + 4t² k, where 0 ≤ t ≤ 1, we can use the arc length formula:

L = ∫ √[dx/dt)² + (dy/dt)²  + (dz/dt)² ] dt

First, let's find the derivatives of each component of the vector r(t):

dx/dt = 9

dy/dt = (\(8t^{3/2}\))) * (3/2) * (2/\(\sqrt{t}\)) = 12t² /\(\sqrt{t}\) = 12\(t^{(5/2)}\)

dz/dt = 8t

Next, we substitute these derivatives into the arc length formula:

L = ∫ √[(dx/dt)²  + (dy/dt)²  + (dz/dt)² ] dt

L = ∫ √[9²  + (12t^(5/2))²  + 8² ] dt

L = ∫ √[81 + 144t⁵ + 64] dt

L = ∫ √[144t⁵ + 145] dt

Now we integrate with respect to t:

L = ∫  \((144t⁵ + 145)^{(1/2)}\) dt

This integral can be challenging to solve analytically. However, we can approximate the length using numerical methods such as numerical integration techniques or software.

Using numerical integration software, we find that the length of the curve is approximately 34.10 units.

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Answer:

-8 (2+1)

16x - 28

Step-by-step explanation:

The three major chords built on white notes without accidentals are:

1. C major chord (C, E, G)

2. F major chord (F, A, C)

3. G major chord (G, B, D)

These chords are formed by taking the root note, skipping one white note, and adding the next white note on top. For example, in the C major chord, the notes C, E, and G are played together to create a harmonious sound.

Similarly, the F major chord is formed by playing F, A, and C, and the G major chord is formed by playing G, B, and D. These three major chords are commonly used in various musical compositions and are fundamental building blocks in music theory.

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The probability that the sample mean will be between $10 and $10.2 is approximately 0.3144 or 31.44%.

To calculate the probability that the sample mean will be between $10 and $10.2, we need to use the properties of the sampling distribution of the sample mean. Under certain assumptions (e.g., a large sample size or normality of the population distribution), the sampling distribution of the sample mean follows a normal distribution.

Given that the population mean is $10.2 and the population standard deviation is $1.4, we can use the Central Limit Theorem to approximate the distribution of the sample mean. The Central Limit Theorem states that for a sufficiently large sample size, the sampling distribution of the sample mean will be approximately normal, regardless of the shape of the population distribution.

In this case, the sample size is 39, which is considered reasonably large. Therefore, we can approximate the distribution of the sample mean as a normal distribution with a mean of $10.2 and a standard deviation of σ/√n, where σ is the population standard deviation and n is the sample size.

The standard deviation of the sample mean is calculated as follows:

Standard deviation of the sample mean = σ/√n = 1.4/√39 ≈ 0.224

To find the probability that the sample mean will be between $10 and $10.2, we need to calculate the z-scores for both values and then use the standard normal distribution (z-distribution) to find the probability.

The z-score for $10 is calculated as follows:

z1 = (10 - 10.2) / (0.224) ≈ -0.893

The z-score for $10.2 is calculated as follows:

z2 = (10.2 - 10.2) / (0.224) = 0

Now, we can use a standard normal distribution table, calculator, or software to find the probabilities associated with these z-scores.

The probability that the sample mean will be between $10 and $10.2 is equal to the probability of having a z-score between -0.893 and 0.

Using a standard normal distribution table or calculator, the probability can be calculated as follows:

P(-0.893 ≤ Z ≤ 0) ≈ P(Z ≤ 0) - P(Z ≤ -0.893)

Looking up the values in a standard normal distribution table or using a calculator, we find:

P(Z ≤ 0) ≈ 0.5 (probability of being less than or equal to the mean)
P(Z ≤ -0.893) ≈ 0.1856

Therefore,

P(-0.893 ≤ Z ≤ 0) ≈ 0.5 - 0.1856 ≈ 0.3144

Hence, the probability that the sample mean will be between $10 and $10.2 is approximately 0.3144 or 31.44%.

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When the spring is stretched and the distance from point a to point b is 5.3 feet, the value of θ is 53.13  degrees

The distance between point a to point b = 5.3 feet

The length of the top side = 3 feet

Therefore, it will form a right triangle

Here we have to use trigonometric function

Here adjacent side and the hypotenuse of the triangle is given

The trigonometric function that suitable for the given conditions is

cos θ = Adjacent side / Hypotenuse

Substitute the values in the equation

cos θ = 3 / 5

θ = cos^-1(3 / 5)

θ = cos^-1(0.6)

θ = 53.13 degrees

Therefore, the value of θ is 53.13  degrees

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Decay means the value gets smaller.

In order for a value to get smaller you need to multiply by a value less than 1

The first equation uses 0.5 which is less than 1

The second equation uses 0.4 which is less than 1

The third equation does not have a decimal and uses 4, which is greater than 1, so cannot get smaller.

The fourth equation uses 0.5 which is less than 1

The answer would be the 3rd one.

Notice that in the given sequence the difference between the first two terms is 7 so the solution must be in terms of multiples of 7 plus a constant. The only two options that have this characteristic are first and last ones, for the sum to consider the first term the sum must start for r=0.

Answer: last option.

Check the picture below.

so the surface area of the pyramid is really just the area of four triangles with a base of 7 and a height of "h" and a 7x7 square, and we also know that that is 210 yd²

\(\stackrel{ \textit{\LARGE Areas} }{\stackrel{ \textit{four triangles} }{4\left[\cfrac{1}{2}(\underset{b}{7})(h) \right]}~~ + ~~\stackrel{ square }{(7)(7)}}~~ = ~~210\implies 14h+49=210 \\\\\\ 14h=161\implies h=\cfrac{161}{14}\implies h=11.5\)

Answer:

the square pyramid's slope height is roughly 5.75 yards.

Step-by-step explanation:

The square pyramid's slope height should be "l."

A square pyramid's surface size is determined by:

Surface area equals base area plus (1/2) * base radius plus slope height.

Given that the base's length is 7 yards and the surface size is 210 yards. The base's size is thus:

Base area equals side times side, or 7 times 7 square yards.

As a result, we have:

210 = 49 + (1/2) * 4 * 7 * l

When we simplify the previous solution, we get:

210 - 49 = 28l

161 = 28l

l = 161/28

Consequently, the square pyramid's slope height is roughly 5.75 yards. (rounded to two decimal places).

Find the ratio of of the common side:

35/15 = 2 1/3

30 x 2 1/3 = 70

Answer: C) 70

Step-by-step explanation:

as you know, there are 12 months per year.

also, the word "annual" means "per year" (of again 12 months).

each worker gets 7,576 per month.

that means each worker gets an annual salary of

7,576 × 12 = 90,912

the salaries/wages of all 53 workers together in a year are then

90,912 × 53 = 4,818,336

Answer:

b= -8

Step-by-step explanation:

The rate of each car is 55 miles per hour.

In order to find the rate of each car, we need to follow the given steps :
1. Let's denote the rate of each car as R (in miles per hour).

2. The first car traveled north from 10 a.m. to 4 p.m., which is 6 hours. So, the distance covered by the first car can be represented as 6R.

3. T he second car traveled south from 12 p.m. to 4 p.m., which is 4 hours. So, the distance covered by the second car can be represented as 4R.

4. According to the problem, the total distance between the two cars at 4 p.m. is 550 miles. Therefore, the sum of the distances covered by both cars should equal 550 miles.

5. Now, we can set up an equation: 6R + 4R = 550

6. Combine the terms: 10R = 550

7. Solve for R: R = 550 / 10 = 55

So, the rate of each car was 55 miles per hour.

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The value of the area integral \($-2 \int\left(\mathrm{x}^2+\mathrm{y}^2-8\right) \mathrm{dxdy}\) is \(4.288 \pi\)

It is given that,

Consider the lower portion of the sphere \($\mathrm{x}^2+\mathrm{y}^2+\mathrm{z}^2=36$\) by the cone z+12=0 or \(z={\sqrt{12}}\)

Let \(x=rcos \theta\) and \(x=rcos \theta\)

Here, the angle \($\theta$\) lies between 0 to \(\pi\).

Then, r lies between 0 to \(2 \sqrt{2}\)

Expressing the area in integral

\($$\begin{aligned}A =-\int_C f(x, y) d A \\ =-\int_C\left(x^2+y^2-{\sqrt{12}}\right) dxdy \\\end{aligned}$$\)

Finding the integral value

\($$\begin{aligned}\mathrm{A} =\int_0^\pi \int_0^{2 \sqrt{2}}\left(\mathrm{r}^2-{\sqrt12}\right) \mathrm{rdrd} \theta \\= \int_0^\pi \int_0^{2 \sqrt{2}}\left(\mathrm{r}^3-{\sqrt{12}} \mathrm{r}\right) \mathrm{drd} \theta \\ =-2 \int_0^\pi\left[\frac{\mathrm{r}^4}{4}-\frac{{\sqrt{12}} \mathrm{r}^2}{4}\right]_0^{2 \sqrt{2}} \mathrm{drd} \theta \\ =-2 \int_0^\pi \frac{(2 \sqrt{2})^4}{4}-\frac{{\sqrt{12}}(2 \sqrt{2})^2}{4} \mathrm{~d} \theta\end{aligned}\)

Simplifying the area integral

\($$\begin{aligned}A =-2 \int_0^\pi \frac{16(4)}{4}-\frac{{\sqrt{12}}(4)^2}{4} \mathrm{~d} \theta \\=-2 \int_0^\pi(16-{\sqrt{12}}\times 4) \mathrm{d} \theta \\=4.288 \int_0^\pi \mathrm{d} \theta \\=4.288[\theta]_0^\pi \\=4.288\pi\end{aligned}$$\)

Hence, the value of the area integral \(-2 \int\left(\mathrm{x}^2+\mathrm{y}^2-8\right) \mathrm{dxdy}\) is \(4.288 \pi\)

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