Furnace repair bills are normally distributed with a mean of 267 dollars and a standard deviation of 20 dollars. If 64 of these repair bills are randomly selected, find the probability that they have a mean cost between 267 dollars and 269 dollars.

Answer:

a)

Kyle's z-score was 1.9 to the nearest tenth

Ethan's z-score was -0.8 to the nearest tenth

b)

The percent of the students had a final exam score lower than Ethan's score was 21.19%

Step-by-step explanation:

a) Lets revise how to find the z-score

- The rule the z-score is z = (x - μ)/σ , where

# x is the score

# μ is the mean

# σ is the standard deviation

* Lets solve the problem

- The exam scores are normally distributed with a mean of 105 and a

  standard deviation of 16

∴ μ = 105 and σ = 16

- Kyle and Ethan are took the final exam

- Kyle's  score was 135

- Ethan's score was 93

- Lets find the z-score for each one

∵ Kyle's  score was 135

∴ x = 135

∵ μ = 105 and σ = 16

∵ z-score = (x - μ)/σ

∴ z-score for Kyle = (135 - 105)/16 = 30/16 = 15/8 = 1.875

* Kyle's z-score is 1.9 to the nearest tenth

∵ Ethan's  score was 93

∴ x = 93

∵ μ = 105 and σ = 16

∵ z-score = (x - μ)/σ

∴ z-score for Ethan = (93 - 105)/16 = -12/16 = -3/4 = -0.75

* Ethan's z-score is -0.8 to the nearest tenth

b) To find the percent of students with a lower exam score than Ethan

   you will asking to find the proportion of area under the standard

   normal distribution curve for all z-scores < -0.8

- It can be read from a z-score table by referencing a z-score of -0.8

- Look to the attached file

∴ The value from the table is 0.2119

- To change it to percent multiply it by 100%

∴ 0.2119 × 100% = 21.19%

* The percent of the students had a final exam score lower than

  Ethan's score was 21.19%

Kyle's z-score is 1.9, and Ethan's z-score is -0.8. Approximately 21.1% of the students had a final exam score lower than Ethan's score.

For Kyle, the z-score is:

Z = (135 - 105) / 16 = 30 / 16 = 1.875, which rounds to 1.9.

For Ethan, the z-score is:

Z = (93 - 105) / 16 = -12 / 16 = -0.75, which rounds to -0.8.

Ethan's z-score correlates to a percentile that represents the percentage of students with scores lower than his. Consulting a standard normal distribution table or using a calculator that provides cumulative probabilities for the normal distribution, we find that a z-score of -0.8 corresponds to approximately 21.1%.

Therefore, about 21.1% of the students had a final exam score lower than Ethan's score.

Answer with explanation:

Given the differential equation

y''+y=0

The two function let

[tex]y_1= sint -cost[/tex]

[tex]y_2=sint+ cost[/tex]

Differentiate [tex]y_1 and y_2[/tex]

Then we get

[tex]y'_1= cost+sint[/tex]

[tex]y'_2=cost-sint[/tex]

Because [tex]\frac{\mathrm{d} sinx}{\mathrm{d} x} = cosx[/tex]

[tex]\frac{\mathrm{d}cosx }{\mathrm{d}x}= -sinx[/tex]

We find wronskin to prove that the function  is independent/ fundamental function.

w(x)=[tex]\begin{vmatrix} y_1&y_2\\y'_1&y'_2\end{vmatrix}[/tex]

[tex]w(x)=\begin{vmatrix}sint-cost&sint+cost\\cost+sint&cost-sint\end{vmatrix}[/tex]

[tex]w(x)=(sint-cost)(cost-sint)- (sint+cost)(cost+sint)[/tex]

[tex]w(x)=sintcost-sin^2t-cos^2t+sintcost-sintcost-sin^2t-cos^2t-sintcost[/tex]

[tex]w(x)=-sin^2t-cos^2t[/tex]    

[tex]sin^2t+cos^2t=1[/tex]

[tex]w(x)=-2\neq0[/tex]

Hence, the given two function are fundamental set of function on R.

Answer:

Step-by-step explanation:

Part A

This is a combination problem. Order does not matter.

36C6

36!/(30! 6!)

36 * 35 * 34 * 33 * 32 * 31/ 6!

1402410240/6!

1947792

Part B

1 / (36C6)

0.000000513 or

0.0000005

y is (replace "is" with an equal sign) 4 less (replace with subtraction sign) than the product (multiply 5 and x) of 5 and x

y = 5x - 4

The reason the answer is like this ^^^ instead of y = 4 - 5x is because for this to be true it would have to say y is 5x less then 4

Hope this helped!

~Just a girl in love with Shawn Mendes

Answer:

y=5x-4

Step-by-step explanation:

Question about: ⇒ algebraic expression

Y: ⇒ Symbol into letters

is: ⇒ equal sign

less than: ⇒ <

product: ⇒ multiply

y=5x-4 is the correct answer.

I hope this helps you, and have a wonderful day!

Answer:

100

Step-by-step explanation:

The value of the given logarithm is 100.

A logarithm is the power to which a number must be raised in order to get some other number.

Given that, log x = 2,

We will solve a logarithmic equation of x  by changing it to exponential form.

Now, the logarithmic equation is log₁₀x = 2

Since, we know that, logₐb = x then b = aˣ

Therefore, log₁₀x = 2

x = 10²

x = 100

Hence,  the value of the given logarithm is 100.

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ANSWER

The value of this function at x=-1 is 1

EXPLANATION

The given function is

[tex]f(x) = 3x + 4[/tex]

We want to find the value of this function at x=-1.

We substitute x=-1 into the function to obtain:

[tex]f( - 1) = 3( - 1)+ 4[/tex]

We multiply out to obtain:

[tex]f( - 1) = - 3+ 4[/tex]

[tex]f( - 1) = 1[/tex]

Therefore the value of this function at x=-1 is 1.

Answer:  [tex]f(-1)=1[/tex]

Step-by-step explanation:

Given the linear function f(x):

[tex]f(x)=3x+4[/tex]

By definition. a relation is a function if each input value has only one output value. In this case you need to find the output value for the input value [tex]x=-1[/tex]. In order to do this, you need to substitute this value of the variable "x" into the linear function given.

Then:

When [tex]x=-1[/tex]:

[tex]f(-1)=3(-1)+4[/tex]

Remember the multiplication of signs:

[tex](+)(-)=-\\(+)(+)=+\\(-)(-)=+[/tex]

Then, the value of f(x) when [tex]x=-1[/tex] is:

 [tex]f(-1)=-3+4[/tex]

 [tex]f(-1)=1[/tex]

Answer:

3t^2s

Step-by-step explanation:

15/3=5

t^5s/t^2s = t^3

t^2s^3/t^2s = s^2

For this case we have by definition, that a polynomial has a common factor when the same quantity, either number or letter, is found in all the terms of the polynomial.

We have the following expression:

[tex]3t ^ 5s-15t ^ 2s ^ 3[/tex]

So we have to:

[tex]3t ^ 2s[/tex] is the lowest common term in the terms of the expression:

[tex]3t ^ 2s (t ^ 3-5s ^ 2)[/tex]

Answer:

[tex]3t ^ 2s[/tex]

Maximum is the highest a graph can reach. In this case the graph continues forever therefore the maximum is:

infinity or ∞

The minimum is the lowest place the graph reaches. In this case it would be:

-6

The zeros are where the graph intersects the x axis. In this case it would have two zeros, which are:

(-3, 0) and (0.5, 0)

Hope this helped!

~Just a girl in love with Shawn Mendes

c. Reflection over either the x-axis or y-axis will change the graph

This is false because [tex]y=sin(x)[/tex] is an odd function, not an even one. This means that [tex]sin(-x)=-sin(x)[/tex], and a reflection over the y-axis will change the graph.

This is false because we said that [tex]sin(-x)=-sin(x)[/tex]

This is true. Since [tex]sin(x)[/tex] is an odd function, then reflection over either the x-axis or y-axis will change the graph as we said in a. So, for [tex]f(x)[/tex]:

REFLEXION IN THE X-AXIS:

[tex]h(x)=-f(x)[/tex]

REFLEXION IN THE Y-AXIS:

[tex]h(x)=f(-x)[/tex]

False by the same explanation as b.

The correct statement about the function y=sin(x) is that Reflection over either the x-axis or y-axis will change the graph. Therefore, option C is the correct answer.

The statement regarding the function y=sin(x) which is true is that reflection over either the x-axis or y-axis will change the graph.

This is because the sine function is an odd function, meaning that it has rotational symmetry about the origin. A characteristic of odd functions is that they satisfy the identity y(-x) = -y(x), not y(-x) = y(x), which describes an even function.

Therefore, the assumption Sin(x)=Sin(-x) would be incorrect, as it does not reflect the odd nature of the sine function. Thus, the correct answer is c. Reflection over either the x-axis or y-axis will change the graph.

Answer with explanation:

It is given that for three integers , a, b and c, if

              [tex]\frac{ab}{c}\rightarrow then, \frac{a}{c} \text{or} \frac{b}{c}[/tex]

Since , a b is divisible by c , following are the possibilities

1.→ a and b are prime integers .Then , c will be prime number either equal to a or b.

2.→a and b are not prime integers ,then any of the factors of a or b will be equal to c.For example:

 ⇒a=m × n

 b=p × q× c

or,

⇒a=u×v×c

b=s×t

So, whatever the integral values taken by a, and b, if [tex]\frac{ab}{c}[/tex] then either of  [tex]\frac{a}{c} \text{or} \frac{b}{c}[/tex] is true.

Answer:

  102.5 g

Step-by-step explanation:

The mass is the sum of the values indicated by the sliders:

  100 g + 0 g + 2.5 g = 102.5 g

The mass, in grams, of the object being measured in the triple beam balance shown below is:

                               102.5 g

Triple Beam balance--

It is a instrument which is used to measure the mass of an object.

The advantage of using this device is that it measures the mass of an object precisely.

It has three counterweights in it.

One is of 100 gram, other is of 10 gram and the last is of 1 gram.

In order to find the mass of an object we add the weights in all the three sections.

The large slider is at 100 g, the medium slider is at 0 g, and the small slider is at 2.5 g.

                    100 g+0 g+2.5 g=102.5 g

Answer:

The height of ball is 17 ft at t=0.55 and t=1.14.

Step-by-step explanation:

The general projectile motion is defined as

[tex]y=-16t^2+vt+y_0[/tex]

Where, v is initial velocity and y₀ is initial height.

It is given that the initial height is 7 and the initial upward velocity is 27.

Substitute v=27 and y₀=7 in the above equation to find the model for height of the ball.

[tex]h(t)=-16t^2+27t+7[/tex]

The height of ball is 17 ft. Put h(t)=17.

[tex]17=-16t^2+27t+7[/tex]

[tex]0=-16t^2+27t-10[/tex]

On solving this equation using graphing calculator we get

[tex]t=0.549,1.139[/tex]

[tex]t\approx 0.55,1.14[/tex]

Therefore the height of ball is 17 ft at t=0.55 and t=1.14.

Answer:

a) The expected value of M = 40

The standard error for M = 4

b) The expected value of M = 40

The standard error for M = 2

Step-by-step explanation:

* Lets revise some definition to solve the problem

- The mean of the distribution of sample means is called the expected

  value of M

- It is equal to the population mean μ

- The standard deviation of the distribution of sample means is called

  the standard error of M

- The rule of standard error is σM = σ/√n , where σ is the standard

  deviation and n is the size of the sample

* lets solve the problem

- A sample is selected from a population

∵ The mean of the population μ = 40

∵ The standard deviation σ = 8

a) The sample has n = 4 scores

∵ The expected value of M = μ

∵ μ = 40

∴ The expected value of M = 40

∵ The standard error of M = σ/√n

∵ σ = 8 and n = 4

∴ σM = 8/√4 = 8/2 = 4

∴ The standard error for M = 4

b) The sample has n = 16 scores

∵ The expected value of M = μ

∵ μ = 40

∴ The expected value of M = 40

∵ The standard error of M = σ/√n

∵ σ = 8 and n = 16

∴ σM = 8/√16 = 8/4 = 2

∴ The standard error for M = 2

When the sample has n = 4 scores then the expected value of M is 40 and the standard error of M is 4.

When the sample has n = 16 scores then the expected value of M is 40 and the standard error of M is 2.

Given

A sample is selected from a population with a mean of μ = 40 and a standard deviation of σ = 8. a. If the sample has n = 4 scores.

The mean of the distribution of sample means is called the expected value of M.

The standard deviation of the distribution of sample means is called the standard error of M.

1. The sample has n = 4 scores

The expected value of M = μ

The expected value of M = 40

The standard error of M is;

[tex]\rm Standard \ error=\dfrac{\sigma}{\sqrt{n} }\\\\ \sigma = 8 \ and \ n = 4}\\\\ Standard \ error=\dfrac{8}{\sqrt{4}}\\\\ Standard \ error=\dfrac{8}{2}\\\\ Standard \ error=4[/tex]

The standard error for M = 4

2.  1. The sample has n = 16 scores

The expected value of M = μ

The expected value of M = 40

The standard error of M is;

[tex]\rm Standard \ error=\dfrac{\sigma}{\sqrt{n} }\\\\ \sigma = 8 \ and \ n = 16}\\\\ Standard \ error=\dfrac{8}{\sqrt{16}}\\\\ Standard \ error=\dfrac{8}{4}\\\\ Standard \ error=2[/tex]

The standard error for M = 2

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Answer: $64

Step-by-step explanation:

Set up is/of ratio. See photo attached. (:

Answer:

The solution is [tex]y(t)=e^{-t}(\cos 32t + (\frac{5}{8}) \sin 32t)[/tex]

Step-by-step explanation:

We need to find the solution of [tex]y''+2y'+17y=0[/tex] with

condition [tex]y(0)=3,\ y'(0)=17[/tex]

This is a homogeneous equation with characteristic polynomial

[tex]r^{2}+2r+17=0[/tex]

using quadratic formula [tex]x=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}[/tex]

[tex]r=\frac{-2\pm \sqrt{2^{2}-4(1)(17)}}{2(1)}[/tex]

[tex]r=\frac{-2\pm \sqrt{4-68}}{2}[/tex]

[tex]r=\frac{-2\pm \sqrt{-64}}{2}[/tex]

[tex]r=\frac{-2\pm 64i}{2}[/tex]

[tex]r=-1 \pm 32i[/tex]

The general solution for eigen value [tex]a \pm ib[/tex] is

[tex]y(t)=e^{at}(A \cos bt + B \sin bt)[/tex]

[tex]y(t)=e^{-t}(A \cos 32t + B \sin 32t)[/tex]

Differentiate above with respect to 't'

[tex]y'(t)=-e^{-t}(A \cos 32t + B \sin 32t) + e^{-t}(-32A \sin 32t + 32B \cos 32t)[/tex]

Since, y(0)=3

[tex]y(0)=e^{0}(A \cos(0) + B \sin(0))[/tex]

[tex]3=(A \cos(0) +0)[/tex]

so, A=1

Since, y'(0)=17

[tex]y'(0)=-e^{0}(3 \cos(0) + B \sin(0)) + e^{0}(-32(3) \sin(0) + 32B \cos (0))[/tex]

[tex]17=-(3 \cos(0)) + (0 + 32B \cos (0))[/tex]

[tex]17=-3 + 32B[/tex]

add both the sides by 3,

[tex]17+3 = 32B[/tex]

[tex]20= 32B[/tex]

divide both the sides, by 32,

[tex]\frac{20}{32}= B[/tex]

[tex]\frac{5}{8}= B[/tex]

Put the value of constants in [tex]y(t)=e^{-t}(A \cos 32t + B \sin 32t)[/tex]

[tex]y(t)=e^{-t}((1) \cos 32t + (\frac{5}{8}) \sin 32t)[/tex]

Therefore, the solution is [tex]y(t)=e^{-t}(\cos 32t + (\frac{5}{8}) \sin 32t)[/tex]

The simplified value is (x  - 6)

[tex]\frac{x^{2} -3x-18}{x+3}\\ = \frac{x^{2} -6x + 3x - 18}{x+3} \\=\frac{x(x-6) +3(x-6)}{x+3}\\ =\frac{(x+3)(x-6)}{x+3}\\ = x - 6[/tex]

So the simplified value is (x  - 6)

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To simplify the rational expression x^2 - 3x - 18 / x + 3, factor the numerator and cancel out the common factor (x + 3). The simplified form is x - 6.

Simplifying the Rational Expression

To simplify the expression
x² - 3x - 18 / x + 3, follow these steps:

(x - 6)(x + 3) / x + 3.

Next, cancel the common factor (x + 3):

(x - 6) (x + 3) / (x + 3) = x - 6

So, the simplified form of the expression is x - 6. Note that this simplification is valid for all values of x except -3, as the denominator would be zero.

ANSWER

The z-score is 2.

EXPLANATION

The z-score for a data set that is normally distributed is calculated using the formula:

[tex]z = \frac{x - \bar x}{ \sigma} [/tex]

where

[tex]\bar x[/tex]

is the mean and

[tex] \sigma[/tex]

is the standard deviation of the distribution.

From the given information the test score is 85.

This implies that,

[tex]x = 85[/tex]

The mean is 75.

[tex]\bar x = 75[/tex]

The standard deviation is 5.

We substitute the values into the formula to get,

[tex]z = \frac{85 - 75}{5} [/tex]

This implies that

[tex]z = \frac{10}{5} [/tex]

Therefore the z-score is

[tex]z = 2[/tex]

The z score for a score of 85 on the test is 2, indicating that it is 2 standard deviations above the mean.

To compute the z score value for a score of 85 on a test with a mean of 75 and a standard deviation of 5, we use the formula:

z = (x - μ) / σ

where z is the z score, x is the score, μ is the mean, and σ is the standard deviation.

Plugging in the values, we get:

z = (85 - 75) / 5 = 2

The z score for a score of 85 is 2. This means the score is 2 standard deviations above the mean.

Answer:

a) $40920

b) $38500

Step-by-step explanation:

Given:

5 employees,

Mean = $40300

Median = $38500

Min = $32000

If he lowest paid employee gets a $3100 raise, then his salary becomes

$32000+$3100=$35100

a) If the mean was $40300, then the sum of 5 salaries is

[tex]\$40300\cdot 5=\$201500[/tex]

After raising the lowest salary the sum becomes

[tex]\$201500+\$3100=\$204600[/tex]

and new mean is

[tex]\dfrac{\$204600}{5}=\$40920[/tex]

b) The  lowest salary becomes $35100. It is still smaller than the median, so the new median is the same as the old one.

New median = $38500

The values of x for which the given vectors are basis for R³ is:

                        [tex]x\neq 1[/tex]

We know that for a set of vectors are linearly independent if the matrix formed by these set of vectors is non-singular i.e. the determinant of the matrix formed by these vectors is non-zero.

We are given three vectors as:

(-1,0,-1), (2,1,2), (1,1, x)

The matrix formed by these vectors is:

[tex]\left[\begin{array}{ccc}-1&2&1\\0&1&1\\-1&2&x\end{array}\right][/tex]

Now, the determinant of this matrix is:

[tex]\begin{vmatrix}-1 &2 & 1\\ 0& 1 & 1\\ -1 & 2 & x\end{vmatrix}=-1(x-2)-2(1)+1\\\\\\\begin{vmatrix}-1 &2 & 1\\ 0& 1 & 1\\ -1 & 2 & x\end{vmatrix}=-x+2-2+1\\\\\\\begin{vmatrix}-1 &2 & 1\\ 0& 1 & 1\\ -1 & 2 & x\end{vmatrix}=-x+1[/tex]

Hence,

[tex]-x+1\neq 0\\\\\\i.e.\\\\\\x\neq 1[/tex]

[tex]|\Omega|=2^4=16\\|A|=4\\\\P(A)=\dfrac{4}{16}=\dfrac{1}{4}[/tex]

Answer:  y = -2x + 13

Step-by-step explanation:

Parallel lines have the same slope.  y = -2x - 2 has a slope of -2 so the line parallel to that will also have a slope of -2.

We have a point (4, 5) and a slope (-2) so we can use the point-slope formula:

y - y₁ = m(x - x₁)    ; where (x₁, y₁) is the point and m is the slope

y - 5 = -2(x - 4)

y - 5 = -2x + 8

y      = -2x + 8 + 5

y      = -2x + 13

Answer:

The equation of a line parallel to line CD is y = -2x + 13 ⇒ 1st answer

Step-by-step explanation:

* Lets revise the conditions of the parallel lines

- The slopes of the parallel lines are equal

- The form of slope-intercept equation is y = m x + c, where

 m is the slope of the line and c is the y-intercept

- The y-intercept means that the line intersect the y-axis at point (0 , c)

- To find an equation of a line parallel to another line, do these steps

# Find the slope of the given line and use it as a slope of the new line

# Substitute x and y in the equation by a point on the new line to find c

* Lets solve the problem

∵ The equation of line CD is y = -2x - 2

∵ The equation of any line is y = m x + c, where m is the slope of

  the line

∴ The slope of the line is -2

- The equation of the line parallel to CD will have the same slope

∵ The parallel line have same slopes

∴ The slope of the new line is -2

∴ The equation of the parallel line is y = -2x + c

- To find c use a point on the new line and replace x and y in the

  equation by its coordinates

∵ The parallel line contains point (4 , 5)

- Put y = 5 and x = 4 in the equation

∴ 5 = -2(4) + c ⇒ simplify

∴ 5 = -8 + c ⇒ add 8 to both sides

∴ 13 = c

- Write the equation with the value of c

∴ y = -2x + 13

* The equation of a line parallel to line CD is y = -2x + 13

Answer: 0.8413

Step-by-step explanation:

Given: Mean : [tex]\mu=\$40,500[/tex]

Standard deviation : [tex]\sigma = \$4,500[/tex]

The formula to calculate z-score is given by :_

[tex]z=\dfrac{x-\mu}{\sigma}[/tex]

For x= $36,000.00, we have

[tex]z=\dfrac{36000-40500}{4500}=-1[/tex]

The P-value = [tex]P(z\geq-1)=1-P(z<-1)=1-0.1586553=0.8413447\approx0.8413[/tex]

Hence, the probability that a randomly selected individual with an undergraduate business degree will get a starting salary of at least $36,000.00 = 0.8413

The probability of a randomly selected individual with an undergraduate business degree having a starting salary of at least $36,000, based on the given normal distribution with a mean of $40,500 and a standard deviation of $4,500, is approximately 0.8413 or 84.13%.

The question asks us to find the probability that a randomly selected individual with an undergraduate business degree will have a starting salary of at least $36,000.00, given that the mean starting salary is $40,500 with a standard deviation of $4,500. This problem can be solved using the properties of the normal distribution.

First, we calculate the z-score, which is the number of standard deviations away from the mean:

Z = (X - μ) / σ

Where X is the salary in question ($36,000), μ is the mean ($40,500), and σ is the standard deviation ($4,500). Plugging in the values:

Z = ($36,000 - $40,500) / $4,500 = -1

The next step is to look up this z-score in a standard normal distribution table or use a calculator with a standard normal distribution function to find the area to the right of this z-score. This area represents the probability we are looking for. Let's assume we found this area to be approximately 0.8413.

Therefore, the probability that a randomly selected individual with an undergraduate business degree will have a starting salary of at least $36,000 is about 0.8413 or 84.13%.

The probability that it does not rain either today or tomorrow, based on the given independent probabilities of it raining each day, is 18%.

This problem pertains to the concept of probability in mathematics. Specifically, it revolves around calculating the probability of compound independent events.

First, we need to determine the probability of not raining each day. If there's a 70% chance of rain today, that means there's a 30% chance (100% - 70%) of no rain today. Similarly, if there's 40% chance of rain tomorrow, there's a 60% chance (100% - 40%) of no rain tomorrow.

Given that the probability it rains on each day is independent, we multiply these probabilities together to get our answer. So, the probability that it does not rain either today or tomorrow is 30% * 60% = 18%.

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The probability that there will be no rain today or tomorrow is 0.18.

The probability that there will be no rain today or tomorrow is given by the formula for the union of two independent events:

[tex]\[ P(\text{no rain today or tomorrow}) = P(\text{no rain today}) \times P(\text{no rain tomorrow}) \][/tex]

First, we need to find the probability of no rain on each day. Since the probability of rain is given, we can subtract this from 1 to find the probability of no rain:

[tex]\[ P(\text{no rain today}) = 1 - P(\text{rain today}) \][/tex]

[tex]\[ P(\text{no rain today}) = 1 - 0.70 \][/tex]

[tex]\[ P(\text{no rain today}) = 0.30 \][/tex]

Similarly, for tomorrow:

[tex]\[ P(\text{no rain tomorrow}) = 1 - P(\text{rain tomorrow}) \][/tex]

[tex]\[ P(\text{no rain tomorrow}) = 1 - 0.40 \][/tex]

[tex]\[ P(\text{no rain tomorrow}) = 0.60 \][/tex]

Now, we can calculate the probability of no rain on either day:

[tex]\[ P(\text{no rain today or tomorrow}) = P(\text{no rain today}) \times P(\text{no rain tomorrow}) \][/tex]

[tex]\[ P(\text{no rain today or tomorrow}) = 0.30 \times 0.60 \][/tex]

[tex]\[ P(\text{no rain today or tomorrow}) = 0.18 \][/tex]

[tex]x\dfrac{\mathrm dy}{\mathrm dx}-y=x^2\sin x[/tex]

Divide both sides by [tex]x^2[/tex]. In doing so, we force any possible solutions to exist on either [tex](-\infty,0)[/tex] or [tex]\boxed{(0,\infty)}[/tex] (the "positive" interval in such a situation is usually taken over the "negative" one) because [tex]x[/tex] cannot be 0 in order for us to do this.

[tex]\dfrac1x\dfrac{\mathrm dy}{\mathrm dx}-\dfrac1{x^2}y=\sin x[/tex]

Condense the left side as the derivative of a product, then integrate both sides and solve for [tex]y[/tex]:

[tex]\dfrac{\mathrm d}{\mathrm dx}\left[\dfrac yx\right]=\sin x[/tex]

[tex]\dfrac yx=\displaystyle\int\sin x\,\mathrm dx[/tex]

[tex]\boxed{y=Cx-x\cos x}[/tex]

The general solution of a differential equation is to write y as a function of x.

Given

[tex]x \frac{dy}{dx} - y = x^2 \sin(x)[/tex]

Divide through by x

[tex]\frac{x}{x} \frac{dy}{dx} -\frac{y}{x} = \frac{x^2}{x} \sin(x)[/tex]

[tex]\frac{dy}{dx} -\frac{y}{x} = x \sin(x)[/tex]

Let P be function of x. Such that:

[tex]P(x) = -\frac 1x[/tex]

So, we have:

[tex]\frac{dy}{dx} +yP(x) = x\sin(x)[/tex]

Calculate the integrating factor I(x).

So, we have:

[tex]I(x) = e^{\int P(x) dx[/tex]

Substitute [tex]P(x) = -\frac 1x[/tex]

[tex]I(x) = e^{\int-\frac 1x dx[/tex]

Rewrite as:

[tex]I(x) = e^{-\int\frac 1x dx[/tex]

Integrate

[tex]I(x) = e^{-\ln(x)[/tex]

[tex]I(x) = \frac 1x[/tex]

So, we have:

[tex]\frac{dy}{dx} -\frac{y}{x} = x \sin(x)[/tex]

[tex][\frac{dy}{dx} -\frac{y}{x}] \frac 1x = [x \sin(x)] \frac 1x[/tex]

[tex][\frac{dy}{dx} -\frac{y}{x}] \frac 1x =\sin(x)[/tex]

Introduce [tex]I(x) = \frac 1x[/tex].

So, we have:

[tex]\frac{d}{dx}(\frac yx) = \sin(x)[/tex]

Multiply both sides by dx

[tex]d(\frac yx) = \sin(x)\ dx[/tex]

Integrate with respect to x

[tex]\frac yx = -\cos(x) + c[/tex]

Multiply through by x

[tex]y = -x\cos(x) + cx[/tex]

So, the general solution is: [tex]y = -x\cos(x) + cx[/tex], and the interval is [tex](0, \infty)[/tex]

Read more about general solution of a differential equation at:

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Answer:

x < 33.84

Step-by-step explanation:

we have

13.48x-200 < 256.12

Solve for x

Adds 200 both sides

13.48x-200 +200 < 256.12+200

13.48x < 456.12

Divide by 13.48 both sides

13.48x/13.48 < 456.12/13.48

x < 33.84

The solution is the interval ----> (-∞, 33.84)

All real numbers less than 33.84

for this case we have that by definition, the sum of the internal angles of a traingule is 180 degrees.

In addition, the angle "R" of the triangle is given by:

[tex]R = 180-45x[/tex]

So, we have to:

[tex](180-45x) + 25x + (57 + x) = 180\\180-45x + 25x + 57 + x = 180\\-45x + 25x + 57 + x = 0\\-45x + 25x + x = -57\\-19x = -57\\x = \frac {57} {19}\\x = 3[/tex]

Answer:

[tex]x = 3[/tex]

Answer:

0,008 or 0,8%

Step-by-step explanation:

To calculate the probability the selected password is made out only of lower-case letters, if it's only letters, we have first to find out how many passwords could be formed with only letters and with only lower-case letters.

For lowercase letters, we can make this many passwords, since for each of the 7  characters, we can pick among 26 lowercase letters:

NLL = 26 * 26 * 26 * 26 * 26 * 26 * 26

In the same fashion, for the number of passwords consisting only of letters, we can pick among 52 letters for each each character (26 lower-case, 26 upper-case):

NOL = 52 * 52 * 52 * 52 * 52 * 52 * 52

We can rewrite NOL differently to ease our calculations:

NOL = (2 * 26) * (2 * 26) * (2 * 26) * (2 * 26) * (2 * 26) * (2 * 26) * (2 * 26)

or

NOL = 26 * 26 * 26 * 26 * 26 * 26 * 26 * 2 * 2 * 2 * 2 * 2 * 2 * 2

Now we have to find out the probability a password containing only letters (NOL) is a password containing only lowercase letters (NLL).  So, we divide NLL by NOL:

[tex]\frac{NLL}{NOL} = \frac{26 * 26 * 26 * 26 * 26 * 26 * 26}{26 * 26 * 26 * 26 * 26 * 26 * 26 * 2 * 2 * 2 * 2 * 2 * 2 * 2}  = \frac{1}{2 * 2 * 2 * 2 * 2 * 2 * 2} = \frac{1}{2^{7} }[/tex]

The probability is thus 1/2^7 or 1/128 or 0,0078125

Which we are asked to round to 3 decimals... so 0,008 or 0,8%

Answer:

Vertical Angles are equal.  Therefore:

(7x -4) = (6x +8)

x = 12

answer is D

Step-by-step explanation:

Answer:

D) 12

Step-by-step explanation:

The two angles are vertical angles.  Vertical angles are equal

7x-4 = 6x+8

Subtract 6x from each side

7x-6x -4 = 6x-6x+8

x-4 = 8

Add 4 to each side

x-4+4 = 8+4

x = 12

Answer: [tex]Y=-388+135X[/tex]

Step-by-step explanation:

The equation of the regression line has the general form [tex]Y=a+bX[/tex], where Y is the dependent variable , X is the independent variable , b is the slope of the line and a is the y-intercept.

Given : The slope of regression equation : [tex]b=135[/tex]

The y-intercept : [tex]a=-388[/tex]

Then , the equation of the regression line is given by :-

[tex]Y=-388+135X[/tex]

In the field of statistics, specifically regression analysis, the equation of the regression line is given by y = mx + b. Given that the slope is 135 and the y-intercept is -388, the equation of the regression line which represents the relationship between hotel ratings and their prices is y = 135x - 388.

The given problem falls under the branch of statistics, specifically under regression analysis. In a regression equation, x is the independent variable and y is the dependent variable. The equation of a regression line can be expressed in the format y = mx + b, where m represents the slope of the line, and b is the y-intercept.

In the given instance, the slope (m) is 135, and the y-intercept (b) is -388. Therefore, the equation of the regression line which represents the relationship between hotel ratings (x) and their prices (y) is y = 135x - 388.

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