The sides and hypotenuse of a right triangle are strictly increasing with time. At the instant when x is 24 inches and y is 32 inches, dy/dt = 2 dx/dt. If dθ/dt = −0.01 radians per minute at the same instant, what is the value of dy/dt at that same instant ?

Answer:

17,101,185, 269,.... is the solution.

i.e. x≡17 mod(84) is the solution

Step-by-step explanation:

Given that the system is

[tex]x ≡ 2 ( mod 3 )x ≡ 1 ( mod 4 )x ≡ 3 ( mod 7 )[/tex]

Considering from the last as 7 is big,

possible solutions would be 10,17,24,...

Since this should also be 1(mod4) we get this as 1,5,9,...17, ...

Together possible solutions would be 17, 45,73,121,....

Now consider I equation and then possible solutions are

5,8,11,14,17,20,23,26,29,...,47,....75, ....

Hence solution is 17.

Next number satisfying this would be 101, 185, ...

Answer:

The LCD is 72; the sum is 35/72

Step-by-step explanation:

Let's find the least common denominator (LCD) and find a solution.

The given expression: [tex]\frac{1}{3}-\frac{1}{8}+\frac{5}{18}[/tex] has three fractions from which their denominators can be expressed as the multiplication of prime numbers:

Fraction 1: 1/3 --> 3 is a prime number

Fraction 2: 1/8 --> 8=2*4=2*2*2

Fraction 3: 5/18 --> 18=3*6=3*2*3

Now, the next step is considering that if a number is repeated using two different fractions, one of the numbers is deleted. Notice that 'fraction 1' has a 3 and 'fraction 3' also has a 3, so we delete one '3'. Now notice that 'fraction 2' has a 2 and 'fraction 3' also has a 2, so we delete one '2'. So initially we have:

(3)*(2*2*2)*(3*2*3)

But after the previous process (erasing one '3' from the first fraction and one '2' from the second fraction) we now have:

(2*2)*(3*2*3)

Doing the math we obtain (2*2)*(3*2*3)=72, so 72 is our LCD.

Now we have to multiply each fraction in order to obtain the same denominator (LCD=72) for all fractions, so:

For fraction 1: 1/3 --> (1/3)*(24*24)=24/72

For fraction 2: 1/8 --> (1/8)*(9/9)=9/72

For fraction 3: 5/18 --> (5/18)*(4/4)=20/72

Now we can sum all the fractions (remember the correct sign for each fraction):

24/72 - 9/72 + 20/72 = (24-9+20)/72 = 35/72

Answer:

Domain {-1,0,6,10}

Range: {3}

Yes it's a function.

Step-by-step explanation:

Domain is all the x's used by your relation. In case of a set of points all you have to do is your list x's. Domain: {-1,0,6 ,10}.

Range is all the y's being used by your relation. In case of a set of points all you have to do is list your y's. Range {3}.

Function: For it be a function no x can be paired with more than one y. Basically all the x's have to be different. In they are in this case so it is a function.

Answer:

See below.

Step-by-step explanation:

The  domain is the set of x-values = {0, -1, 6, 10}.

The range is {3}.

The relation is a function because each element of the domain maps on to only one value of the range.

Answer:  The required function is R =  {(a, blue), (b, green), (c, green), (d, white), (e, black)}, domain, D = {a, b, c, d, e} and range, R = {blue, green, white, black} .

Step-by-step explanation: We are given the following two sets :

A = {a, b, c, d, e}

and

B = {yellow, orange, blue, green, white, red, black}.

We are to define a relation R from A to B that is a function and contains at least 4 ordered pairs. Also, to find the domain and range of the function.

(a) Let the function R be defined as follows :

R = {(a, blue), (b, green), (c, green), (d, white), (e, black)}.

Since R contains five ordered pairs, so this will fulfill our criterion. Also, since each first element is associated with one and only one second element, so R defines a function.

(b) We know that

the domain of a function is the set of all the first elements in the ordered pairs, so the domain of the function R will be

D = {a, b, c, d, e}.

(c) We know that

the range of a function is the set of all the second elements in the ordered pairs, so the domain of the function R will be

R = {blue, green, white, black}.

Thus, the required function is R =  {(a, blue), (b, green), (c, green), (d, white), (e, black)}, domain, D = {a, b, c, d, e} and range, R = {blue, green, white, black} .

Answer:8

Step-by-step explanation:

We have given test consists of 40 multiple choice questions having five options for each question.

Suppose there is one correct answer to each question

therefore probabilty of getting a correct answer on making a guess is [tex]\frac{1}{5}[/tex]

i.e. 1 out of 5 questions is correct

Using binomial distribution

where n=40 p=[tex]\frac{1}{5}[/tex]

mean of binomial distribution is np

therefore mean of no of correct answers=[tex]40\times \frac{1}{5}[/tex]

                                                                    =8

Answer:

2670 square feet. Option e.

Step-by-step explanation:

Dan and Betty can paint 2,400 square feet in 4 hours.

They can paint in one hour [tex]\frac{2400}{4}[/tex] = 600 square feet.

Since given that Betty paints twice as fast as Dan. Let us take an equation:

Let Betty = B, Dan = D and Sue = S

B = 2D

4(B+D) = 2400

4B + 4D = 2400

12D = 2400

D = 200 sq. ft.

B = 2D = 400 sq. ft.

Therefore, Dan can paint 200 square feet in 1 hour and Betty paints twice 400 square feet in 1 hour.

Now given three of them can paint 3,600 square feet in 3 hours.

3( B+D+S) = 3600

3B + 3D + 3S = 3600

3(400) + 3(200) + 3(S) = 3600

1200 + 600 + 3S = 3600

S = 600 Sq. ft.

Sue can paint 600 square feet in one hour.

So sue can paint in 4 hours and 27 minutes.

[tex](\frac{4+27}{60})[/tex] × 600

= 2670 square feet. Option e.

Sue's painting rate is 600 square feet per hour. For a total of 4 hours and 27 minutes, which is 4.45 hours when converted, she can paint 2,670 square feet.

The question is asking for Sue's rate of painting when she works alone given the painting rates when they all work together. From the given information, we know that Dan and Betty together can paint 2,400 square feet in 4 hours, which means their combined painting rate is 2,400 ÷ 4 = 600 square feet per hour. Additionally, we know that Dan, Betty, and Sue together can paint 3,600 square feet in 3 hours. This means their combined rate is 3,600 ÷ 3 = 1,200 square feet per hour.

Because Sue's rate is the only variable that changes between these two situations, we can determine her rate by subtracting the combined rate of Dan and Betty from the combined rate of the whole team. This gives us 1,200 - 600 = 600 square feet per hour for Sue.

To find out how many square feet she can paint in 4 hours and 27 minutes, we need to convert 27 minutes into hours, which is 27 ÷ 60 = 0.45 hours. Adding this to the 4 hours, we get 4.45 hours. Multiplying Sue's rate by this time gives us 600 × 4.45 = 2,670 square feet, which matches answer option (e).

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Final answer:

Mrs. Alford invested $3437.50 at a 1% interest rate and $3262.50 at a 9% interest rate. The solution involved setting up and solving a system of linear equations based on the given total investment and income.

Explanation:

The problem involves solving a system of linear equations to determine how much money Mrs. Alford invested at 1% and 9%. Let 'x' represent the amount invested at 1% and 'y' represent the amount invested at 9%. The total amount invested is $6700, so the first equation is x + y = 6700. The total annual income from these investments is $275. The income from the investment at 1% is 0.01x, and the income from the investment at 9% is 0.09y, creating the second equation: 0.01x + 0.09y = 275.

To solve the system, we can start by multiplying the second equation by 100 to get rid of the decimals, resulting in 1x + 9y = 27500. Subtracting the first equation from this gives 8y = 26100, which implies that y = 3262.5. Therefore, Mrs. Alford invested $3262.50 at 9%. Using the first equation, we find that x = 6700 - 3262.5 = 3437.5, meaning $3437.50 was invested at 1%.

Answer:

a) 5+13k  where k is integer

b) 20+13k where k is integer

c)12+13k where k is integer

Step-by-step explanation:

(a)

[tex]8x \equiv 1 (mod 13) \text{ means } 8x-1=13k[/tex].

8x-1=13k

Subtract 13k on both sides:

8x-13k-1=0

Add 1 on both sides:

8x-13k=1

I'm going to use Euclidean Algorithm.

13=8(1)+5

8=5(1)+3

5=3(1)+2

3=2(1)+1

Now backwards through the equations:

3-2=1

3-(5-3)=1

3-5+3=1

(8-5)-5+(8-5)=1

2(8)-3(5)=1

2(8)-3(13-8)=1

5(8)-3(13)=1

So compare this to:

8x-13k=1

We see that x is 5 while k is 3.

Anyways 5 is a solution or 5+13k is a solution where k is an integer.

b)

[tex]8x \equiv 4 (mod 13)[/tex]

8x-4=13k

Subtract 13k on both sides:

8x-13k-4=0

Add 4 on both sides:

8x-13k=4

We got this from above:

5(8)-3(13)=1

If we multiply both sides by 4 we get:

8(20)-13(12)=4

So x=20 and 20+13k is also a solution where k is an integer.

c)

[tex]99x \equiv 5 (mod 13)[/tex

99x-5=13k

Subtract 13k on both sides:

99x-13k-5=0

Add 5 on both sides:

99x-13k=5

Using Euclidean Algorithm:

99=13(7)+8

13=8(1)+5

Go back through the equations:

13-8=5

13-(99-13(7))=5

8(13)-99=5

99(-1)+8(13)=5

Compare this to 99x-13k=5 and see that x=-1 or -1+13=12 or 12+13k is a solution where k is an integer.

Answer:

a) x = 5 mod 13.

b)  x = 7 mod 13.

Step-by-step explanation:

a) 8x = 1  mod  13

x = 2,  16 = 3 mod 13

x = 3, 24 = 11 mod 13

x = 4, 32 = 6 mod 13

x = 5 , 40 = 1 mod 13

8x = 40

x = 5 mod 13.

b)   8x = 4 mod 13

x = 7,  56 = 4 mod 13.

7 = 4 mod 13

x = 7 mod 13.

Answer:

[tex]f'(x)=3-9x^{2}[/tex] and [tex]f''(x)=-18x[/tex]

Step-by-step explanation:

In order to find the derivatives, first we need to remember that for polynomial functions:

[tex]f'(x)=(x^{n}+x^{m})'= (x^{n})'+(x^{m})'[/tex], as well as that:

[tex]f'(x)= (x^{n})' = n*(x^{n-1})[/tex]

1. First derivative of the function:

[tex]f(x)=9+3x-3x^{3}[/tex]

[tex]f'(x)=(9)'+(3x)'-(3x^{3})'[/tex] using the property [tex]f'(x)= (x^{n})' = n*(x^{n-1})[/tex] then

[tex]f'(x)=3-3*3x^{2}[/tex], remember that the derivative of a constant is equal to 0

[tex]f'(x)=3-9x^{2}[/tex]

2. Second derivative:

[tex]f'(x)=3-9x^{2}[/tex]

[tex]f''(x)=(3-9x^{2})'[/tex] using the property [tex]f'(x)= (x^{n})' = n*(x^{n-1})[/tex] then

[tex]f''(x)=(3)'-(9x^{2})'[/tex]

[tex]f''(x)=-(9*2)x^{1}[/tex]

[tex]f''(x)=-18x[/tex]

In conclusion, [tex]f'(x)=3-9x^{2}[/tex] and [tex]f''(x)=-18x[/tex]

Check the picture below.

The p-value is obtained by measuring how extreme the observed test statistic is compared to what is expected under the null hypothesis. In this case, a p-value less than the significance level (0.05) leads to the rejection of the null hypothesis, indicating that the actual proportion of readers owning the car is different from the reported 55%.

Given that the publisher claims that 55% of their readers own a particular make of car, this would be our null hypothesis (H0: p = 0.55), with the actual percentage of their readers owning a car being the alternate hypothesis (Ha: p ≠ 0.55). The sample found that 46% of 200 readers owned the car, so our sample proportion (p') is 0.46.

Using these values, we can calculate the test statistic (Z), which measures the number of standard deviations p' is from p under the null hypothesis. Once we have our test statistic, we can then determine the p-value.

The observed value of the test statistic falls into the critical region. The p-value is the probability that a test statistic will take on a value as extreme (or more extreme) than the observed value of the test statistic calculated from your sample data. If the p-value is less than or equal to the level of significance, we reject the null hypothesis. From the available reference, it is found that the p-value is less than the significance level of 0.05, and therefore, we would reject the null hypothesis and conclude that the proportion is different from 55% based on this sample data.

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Answer:

There is only one real zero and it is located at x = 1.359

Step-by-step explanation:

After the 4th iteration the solution was repeating the first 3 decimal places.  The formula for Newton's Method is

[tex]x_{n}-\frac{f(x_{n}) }{f'(x_{n}) }[/tex]

If our function is

[tex]f(x)=x^5+x-6[/tex]

then the first derivative is

[tex]f'(x)=5x^4+1[/tex]

I graphed this on my calculator to see where the zero(s) looked like they might be, and saw there was only one real one, somewhere between 1 and 2.  I started with my first guess being x = 1.

When I plugged in a 1 for x, I got a zero of 5/3.  

Plugging in 5/3 and completing the process again gave me 997/687

Plugging in 997/687 and completing the process again gave me 1.36976

Plugging in 1.36976 and completing the process again gave me 1.359454

Plugging in 1.359454 and completing the process again gave me 1.359304

Since we are looking for accuracy to 3 decimal places, there was no need to go further.

Checking the zeros on the calculator graphing program gave me a zero of 1.3593041 which is exactly the same as my 5th iteration!

Newton's Method is absolutely amazing!!!

Using Newton's Method with an initial guess of 1.5, approximations for f(x) = 0 are x ≈ 1.189, close to the actual zeros.

To approximate the zero(s) of the function f(x) = x^5 + x - 6 using Newton's Method, we start with an initial guess, x0. The formula for the iterative step is:

x_(n+1) = x_n - f(x_n) / f'(x_n),

where f'(x) is the derivative of f(x). In this case, f'(x) = 5x^4 + 1.

1. Choose an initial guess, say x0 = 1.5.

2. Iterate using the formula: x_(n+1) = x_n - (x_n^5 + x_n - 6) / (5x_n^4 + 1).

Continue these iterations until two successive approximations differ by less than 0.001. It may take several iterations to reach this level of accuracy.

After reaching a sufficiently accurate result, use a graphing utility to confirm the zero(s) to three decimal places. This will help ensure the accuracy of the approximation from Newton's Method. The actual zeros of the function are approximately x ≈ 1.189, x ≈ -1.187, and x ≈ 1.999. Compare the results from Newton's Method to these values.

For such more questions on initial

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Answer:

cos3x

Step-by-step explanation:

y" - y' + 9y = 3 sin 3x

[tex]D^{2}y-Dy+9y=3 sin3x[/tex]

[tex]y=\frac{3 sin 3x}{(D^{2} -D+9}=3 sin 3x[/tex]

here [tex]D^2[/tex] will be replaced by  [tex]\alpha^2[/tex] where [tex]\alpha[/tex] is coefficient of x

[tex]y=\frac{3 sin 3x}{-3^{2} -D+9}[/tex]

[tex]y=-3\frac{sin 3x}{D}[/tex]

[tex]y=-3\int\ {sin 3x} \, dx[/tex]

[tex]y=-3\frac{cos3x}{-3}[/tex]

y=cos3x

hence Particular solution is cos3x

To maximize the area of the kennel, the fencing should be arranged to form a square or closely resemble a square. The optimal length for each side would be 120/3 = 40 ft. This arrangement would provide the most amount of space for the dogs.

In the problem highlighted, Lacinda has 120 ft of fence available to make a kennel with her house serving as one side of the rectangle. In terms of mathematics, this is an application of optimization in calculus or geometrical considerations for non-calculus level. However, to maximize the area with a given perimeter, a square or a rectangle closest to a square should be constructed.

If the length of the kennel adjacent to the house is x (ft), the length of the other two sides required would be (120 - x) / 2. Therefore, the area of the rectangle (in square feet) can be represented as (120 - x) * x / 2.

To find the maximum area, we would solve this for x. Without using calculus, we would say that for a rectangle, equality of sides (i.e., square) gives the maximum area. Therefore, the optimal length for x would be 120/3 = 40 ft. This arrangement would provide the maximum area for the kennel.

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The length that maximizes the area of Lacinda's rectangular kennel is 60 feet.

Let the length of the side parallel to the house be L and each of the two widths perpendicular to the house be W.

The total fencing used is 120 ft, so we have the equation:

2W + L = 120

We need to express the area A in terms of a single variable:

A = L * W

From the fencing equation, solve for L:

L = 120 - 2W

Substitute into the area equation:

A = (120 - 2W) * W

A = 120W - 2W²

To maximize the area, take the derivative of A with respect to W and set it to zero:

dA/dW = 120 - 4W = 0

Solve for W:

4W = 120

W = 30

Substitute W back into the fencing equation to find L:

L = 120 - 2 * 30 = 60

Thus, the length that maximizes the area of the kennel is 60 feet.

Answer:

The total amount of gas Ms. Smith used on the trip is $25.

Step-by-step explanation:

To solve this problem, we first must figure out how many gallons of gas Ms. Smith used on her trip.  If she drove 700 miles and averaged 35 miles per gallon, if we divide 700 by 35 we can figure out how many gallons she used.

700/35 = 20

Thus, Ms. Smith used 20 gallons on her trip.  Next, to figure out what the cost of the gasoline was, we must multiply the number of gallons (20) by the cost of gasoline per gallon ($1.25).

20 gallons * $1.25/gal = $25

Therefore, the cost of gasoline for the trip was $25.

Hope this helps!

Answer:

2530 has no units

Step-by-step explanation:

In order to understand the units from a linear equation we need to understand the general equation of a line which is:

y=mx+b where:

m=slope of the line

b=y-intercept.

Comparing the given equation with the general line equation, we noticed that 2530 represents the slope of the line.

Since the slope can be obtained by:

m=(y2-y1)/(x2-x1) whatever the units are, the slope is  dimensionless, which means that 2530 has no units.

52 cards  / 4 suits  = 13 cards of each suit.

Theoretically picking a heart would be 13/52 = 1/4 probability.

Experimentally she picked 15 hearts out of 80 total tries. for a 15/80 = 3/16 probability, which is less than the theoretical probability.

1/4 - 3/16 = 1/16

The answer is A.

Answer:

The right answer is A - The theoretical probability of choosing a heart is StartFraction 1 over 16 EndFraction greater than the experimental probability of choosing a heart.

Answer: [tex]142.58\text{squared kg}[[/tex]

Step-by-step explanation:

Answer:

Step-by-step explanation:

Given : A random sample of 10 subjects have weights with a standard deviation of 11.9407 kg

i.e. [tex]\sigma = 11.9407[/tex]

Since we know that the value of variance is the square of standard deviation.

i.e. [tex]\text{Variance}=\sigma^2[/tex]

Therefore, to find the value of variance, we need to find the square of the given standard deviation.

i.e. [tex]\text{Variance}=(11.9407)^2=142.58031649\approx142.58\text{squared kg}[/tex]

Thus, the variance of their​ weights =[tex]142.58\text{squared kg}[[/tex]

Final answer:

The variance of the sample data is 142.58 kg squared, found by squaring the given standard deviation of 11.9407 kg.

Explanation:

The calculation of variance involves squaring the standard deviation. Given a sample with a standard deviation of 11.9407 kg, the variance can be found by squaring this value:

Variance (
s2) = Standard Deviation (
s)2 = 11.9407 kg2

The variance of the sample data is:

142.58 kg2 (this value has been rounded to four decimal places as instructed).

The units for variance are always the square of the units for the original data, hence the variance of weights is expressed in kilograms squared (kg2).

To find out what percent of the new solution is alcohol after water has been added, we need to follow these steps:
Step 1: Calculate the amount of alcohol in the original solution.
The original solution is 20% alcohol and the total volume of the original solution is 25 ounces. To find the amount of alcohol in the original solution, we multiply the total volume by the percentage of alcohol (in decimal form):
Amount of Alcohol = Total Volume * Alcohol Percentage
                 = 25 ounces * 0.20
                 = 5 ounces
So the original solution contains 5 ounces of alcohol.
Step 2: Calculate the new total volume of the solution after adding water.
We add 50 ounces of water to the original 25 ounces of the solution:
New Total Volume = Original Solution Volume + Water Added
                = 25 ounces + 50 ounces
                = 75 ounces
Step 3: Calculate the new percentage of alcohol in the solution.
The amount of alcohol hasn't changed; it's still the original 5 ounces. The percentage of alcohol in the new solution is the amount of alcohol divided by the new total volume:
New Alcohol Percentage = Amount of Alcohol / New Total Volume
                      = 5 ounces / 75 ounces
                      = 0.0667 (approximately)
To express this as a percentage, we multiply by 100:
New Alcohol Percentage = 0.0667 * 100
                      = 6.67% (approximately)
Therefore, after adding 50 ounces of water to the 25-ounce solution that was originally 20% alcohol, the new solution is approximately 6.67% alcohol.

Answer:

  s/2 +17

Step-by-step explanation:

If s represents the number of stamps Emily started with, then the number she had after selling half of them is ...

  s/2

After purchasing 17 more, she had ...

  s/2 +17

Given that

[tex]\vec G(x,y,z)=xy\,\vec\imath+z\,\vec\jmath+3y\,\vec k[/tex]

has a fairly simple curl,

[tex]\nabla\times\vec G(x,y,z)=2\,\vec\imath-x\,\vec k[/tex]

we can take advantage of Stokes' theorem by transforming the line integral of [tex]\vec G[/tex] along the boundary of the square (call it [tex]S[/tex]) to the integral of [tex]\nabla\times\vec G[/tex] over [tex]S[/tex] itself. Parameterize [tex]S[/tex] by

[tex]\vec s(u,v)=u\,\vec\jmath+v\,\vec k[/tex]

with [tex]-\dfrac92\le u\le\dfrac92[/tex] and [tex]-\dfrac92\le v\le\dfrac92[/tex]. Then take the normal vector to [tex]S[/tex] to be

[tex]\vec s_u\times\vec s_v=\vec\imath[/tex]

so that

[tex]\displaystyle\int_{\partial S}\vec G\cdot\mathrm d\vec r=\iint_S(\nabla\times\vec G)\cdot(\vec s_u\times\vec s_v)\,\mathrm du\,\mathrm dv[/tex]

[tex]=\displaystyle\int_{-9/2}^{9/2}\int_{-9/2}^{9/2}(2\,\vec\imath)\cdot(\vec\imath)\,\mathrm du\,\mathrm dv=\boxed{162}[/tex]

We have,the circulation of  [tex]G=xyi+zi+3yk[/tex]  around a square of side 3, centered at the origin, lying in the yz-plane, and oriented counterclockwise when viewed from the positive x-axis is

[tex]\int_{\theta} G.dr=162[/tex]

From the Question we have the equation to be

[tex]G=xyi+zi+3yk[/tex]

Therefore

[tex]\triangle *G=\begin{Bmatrix}i & j & k\\\frac{d}{dx} & \frac{d}{dy} & \frac{d}{dz}\\xy&z&3y\end{Bmatrix}[/tex]

Where

For i

[tex]i(\frac{d}{dy}*3y-(\frac{d}{dz}*xy))\\\\2i[/tex]

For j

[tex]j(\frac{d}{dx}*3y-(\frac{d}{dz}*xy))[/tex]

[tex]0j[/tex]

For z

[tex]z(\frac{d}{dy}*xy-(\frac{d}{dx}*z))[/tex]

[tex]xk[/tex]

Therefore

[tex]\triangle *G=2i-xk[/tex]

Generally considering [tex]\theta[/tex] as the origin

We apply Stoke's Theorem

[tex]\int_{\theta} G.dr=\int_{\theta}(\triangle *G) i ds[/tex]

[tex]\int_{\theta} G.dr=\int_{\theta}(=2i-xk) i ds[/tex]

[tex]\int_{\theta} G.dr=\int2ds[/tex]

[tex]\int_{\theta} G.dr=2*9^2[/tex]

[tex]\int_{\theta} G.dr=162[/tex]

Therefore

The circulation of [tex]G=xyi+zi+3yk[/tex] around a square of side 3, centered at the origin, lying in the yz-plane, and oriented counterclockwise when viewed from the positive x-axis is

[tex]\int_{\theta} G.dr=162[/tex]

Proof:

Let [tex]X \in P(B-A) [/tex]. As we chose [tex]X[/tex] in [tex]P(B-A) [/tex] we know that [tex]X \subseteq B-A[/tex]. Since [tex]B-A \subseteq B[/tex] by transitivity we get:

[tex]X \subseteq B \quad \implies X \in P(B)[/tex].

If [tex]X [/tex] is the empty set, we already have that [tex]X = \emptyset \in P(B) - P(A)[/tex]. But if [tex]X[/tex] is not empty, that means that it can't be subset of [tex]A[/tex], because [tex] X [/tex] is already  subset of [tex]B-A[/tex], and those sets do not share any element. In other words:

[tex]X \subseteq A \cup (B-A) = \emptyset[/tex]

[tex] \Rightarrow X = \emptyset[/tex]

As [tex]X[/tex] can't be subset of [tex]A[/tex], then [tex] X\notin P(A) [/tex]. [tex]X[/tex] was an arbitrary element, and

Thus, [tex] X\in P(B)-P(A) [/tex], where we conclude that

[tex]P(B-A) \subseteq P(B) - P(A)[/tex]

Answer:

The Total of both the sliced turkey and provolone cheese together is $46.49

Step-by-step explanation:

Hello, great question. These types are questions are the beginning steps for learning more advanced Algebraic Equations.

Assuming that the the sliced turkey costs $0.89 / pound (Since it didn't show up correctly in the question) we can create the following equation to solve for the total cost of the order.

[tex]\frac{24.8 lb}{0.89} + \frac{38.2 lb}{2.05} = Total[/tex]

[tex]27.86 + 18.63 = Total[/tex] ....rounded to nearest hundredth

[tex]46.49 = Total[/tex]

So the total of both the sliced turkey and provolone cheese together is $46.49

I hope this answered your question. If you have any more questions feel free to ask away at Brainly.

The final cost for 24.8 pounds of sliced turkey at $1.89 per pound and 38.2 pounds of provolone cheese at $2.05 per pound is $125.18.

To find the total cost of sliced turkey and provolone cheese purchased by the manager at SUBWAY, we need to calculate the cost for each item separately and then sum them up.

Given:

- Sliced turkey:

 - Amount: 24.8 pounds

 - Price per pound: $1.89

- Provolone cheese:

 - Amount: 38.2 pounds

 - Price per pound: $2.05

Calculating the cost of sliced turkey:

[tex]\[ \text{Cost of turkey} = \text{Amount of turkey} \times \text{Price per pound of turkey} \][/tex]

[tex]\[ \text{Cost of turkey} = 24.8 \, \text{pounds} \times \$1.89/\text{pound} \][/tex]

[tex]\[ \text{Cost of turkey} = \$46.87 \][/tex]

Calculating the cost of provolone cheese:

[tex]\[ \text{Cost of provolone cheese} = \text{Amount of cheese} \times \text{Price per pound of cheese} \][/tex]

[tex]\[ \text{Cost of provolone cheese} = 38.2 \, \text{pounds} \times \$2.05/\text{pound} \][/tex]

[tex]\[ \text{Cost of provolone cheese} = \$78.31 \][/tex]

Finding the final cost (total cost):

To find the total cost, we add the cost of turkey and the cost of provolone cheese:

[tex]\[ \text{Total cost} = \$46.87 + \$78.31 \][/tex]

[tex]\[ \text{Total cost} = \$125.18 \][/tex]

The complete question is

A manager at SUBWAY wants to find the total cost of 24.8 pounds of sliced turkey at $1.89 a pound and 38.2 pounds of provolone cheese at $2.05 a pound. Find the final cost.

Answer:

   WXYZ = Ro(180°, (2, -3))(ABCD)

Step-by-step explanation:

A reflection across two perpendicular lines (y=-3, x=2) is equivalent to reflection across their point of intersection. That, in turn, is equivalent to rotation 180° about that point of intersection.

Your double reflection is equivalent to rotation 180° about (2, -3).

Answer:

A+ Rental charges $0.60 per mile.

Rock Bottom Rental charges $70 plus $0.25 per mile.

Let 'x' be the number of miles driven. Let 'y' be the cost of the rental.

The equation for A+ Rental:

y = $0.60x

The equation for Rock Bottom Rental:

y = $0.25x + $70

Answer:

Mean=18.32

Median=20.7

Standard deviation=7.10

Step-by-step explanation:

Mean is the average of the sample size. It is calculated by dividing the sum of all the observations by the number of observations.

[tex]mean=\frac{7.3+11.7+21.7+18.8+23.2+20.7+21.2+10.6}{9}[/tex]

[tex]\Rightarrow mean=18.32[/tex]

Median is the middle value of the observations if the number of observations is odd. If the number of the observations is even then it is the middle value and the next observations average. The values need to be arranged in ascending order.

Therefore the observations become

[tex]7.3, 10.6, 11.7, 18.8, 20.7, 21.2, 21.7, 23.2, 29.7[/tex]

In this case the number of observations is 9 which is odd

Therefore, the median is 20.7 i.e., the fifth observation

[tex]Standard\ deviation=\sqrt \frac{\sum_{i=1}^{N}\left ( x_{i}-\bar{x} \right )^2}{N-1}[/tex]

[tex]where,\ N=Number\ of\ observations\\\bar x=mean\\x_{i}=x_{1}+x_{2}+x_{3}+x_{4}.........x_{N}[/tex]

[tex]\left ({x_{i}}-\bar {x}\right )}^2[/tex]

[tex]\\121.49\\43.85\\11.41\\0.23\\23.79\\5.65\\129.45\\8.28\\59.63\\\sum_{i=1}^{N}\left ({x_{i}}-\bar {x}\right )}^2=403.80\\N-1=9-1\\=8\\\therefore Standard\ deviation=\sqrt {\frac{403.80}{8}}=7.10[/tex]

Standard deviation=7.10

Final answer:

This detailed answer addresses how to find the mean, median, and standard deviation for a given data set in a High School Mathematics context.

Explanation:

Mean: To find the mean, add up all the values and then divide by the total number of values. Add all the values provided: 7.3 + 11.7 + 21.7 + 18.8 + 23.2 + 20.7 + 29.7 + 21.2 + 10.6 = 164.9. Then, divide by 9 (total values) to get a mean of 164.9/9 = 18.32.

Median: To find the median, arrange the data in numerical order and find the middle value. The data in order is: 7.3, 10.6, 11.7, 18.8, 20.7, 21.2, 21.7, 23.2, 29.7. The median is the middle value, which is 20.7.

Standard Deviation: To calculate the standard deviation, you need to find the variance first. Then, take the square root of the variance. Using the given data above, the standard deviation is approximately 6.94.

Answer:

Approximately 59 stacked cups.

Step-by-step explanation:

Given,

Height of a cup = 12.5 cm,

Two stacked cups = 14 cm,

Three stacked cups = 15.5 cm,

........, so on,....

Thus, there is an AP that represents the given situation,

12.5, 14, 15.5,....

First term is, a = 12.5,

Common difference, d = 1.5 cm,

Thus, the height of x cups is,

[tex]h(x) = a+(x-1)d = 12.5 + (x-1)1.5 = 1.5x + 11[/tex]

According to the question,

h(x) = 200

⇒ 1.5x + 11 = 200

⇒ 1.5x = 189

⇒ x = 59.3333333333 ≈ 59,

Hence, approximately 59 stacked cups will need.

Answer:

hx = 1.5cm . x + 11 cm

126 cups

Step-by-step explanation:

We have the following ordered pairs (x, hx).

We are looking for a linear equation of the form:

hx = a.x + b

where,

a is the slope

b is the y-intercept

To find the slope, we take any pair of ordered values and replace their values in the following expression.

[tex]a=\frac{\Delta hx }{\Delta x} =\frac{h2-h1}{2-1} =\frac{14cm-12.5cm}{2-1} =1.5cm[/tex]

Now, the general form is:

hx = 1.5cm . x + b

We can take any ordered pair and replace it in this expression to find b. Let's use h1.

h1 = 1.5cm . x1 + b

12.5 cm = 1.5 cm . 1 + b

b = 11 cm

The final equation is:

hx = 1.5cm . x + 11 cm

If hx = 200 cm,

200 cm = 1.5cm . x + 11 cm

189 cm = 1.5cm . x

x = 126

Answer:

The growth rate is 1.02; the value of the house in 2010 is $185,388

Step-by-step explanation:

This is an exponential growth equation, therefore, it follow the standard form:

[tex]y=a(b)^x[/tex]

where y is the value of the house after a certain number of years,

a is the initial value of the house,

b is the growth rate, and

x is the year number.

We are going to make this easy on ourselves and call year 1985 year 0.  Therefore, is year 1985 is year 0, then year 2005 is year 20, and year 2010 is year 25.  We will make these the x coordinates in our coordinate pairs.

(0, 113000) and (20, 155000)

Filling into our standard form using the first coordinate pair will give us the initial value of the house at the start of our problem:

[tex]113000=a(b)^0[/tex]

Anything raised to the 0 power is equal to 1, so

113000 = a(1) and

a = 113000

Now we will use that value of a along with the second pair of coordinates and solve for b, the growth rate you're looking for:

[tex]155000=113000(b)^{20}[/tex]

Start by dividing both sides by 113000 to get a decimal:

[tex]1.371681416=b^{20}[/tex]

To solve for b, we have to undo that power of 20 by taking the 20th root of b.  Because this is an equation, we have to take the 20th root of both sides:

[tex]\sqrt[20]{1.371681416}=\sqrt[20]{b^{20}}[/tex]

The 20th root and the power of 20 undo each other so all we have left on the right is a b, and taking the 20th root on your calculator of the decimal on the left gives you:

b = 1.0159 which rounds to

b = 1.02  This is our growth rate.

Now we can use this growth rate and the value of a we found to write the model for our situation:

[tex]y=113000(1.02)^x[/tex]

If we want to find the value of the house in the year 2010 (year 25 to us), we sub in a 25 for x and do the math:

[tex]y=113000(1.02)^{25}[/tex]

Raise 1.02 to the 25th power and get:

y = 113000(1.640605994) and multiply to get a final value of

y = $185,388

The annual exponential growth rate from 1985 to 2005 is 1.40%. Using this rate, the estimated value of the house in 2010 would be approximately $176,927.

In order to calculate the exponential growth rate, we use the formula: R = (final value/initial value)^(1/n) - 1, where R is the annual rate, n is the number of years. So, R = (155,000/113,000)^(1/20) - 1 = 0.0140 or 1.40%.

To find the value of the house in 2010, we use the exponential growth formula: future value = present value * (1 + annual rate)^n. The value in 2010 would be $155,000 * (1 + 0.0140)^5 = $176,927 (rounded to the nearest dollar).

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If

[tex]y=\displaystyle\sum_{n=0}^\infty a_nx^n[/tex]

then

[tex]y'=\displaystyle\sum_{n=1}^\infty na_nx^{n-1}=\sum_{n=0}^\infty(n+1)a_{n+1}x^n[/tex]

The ODE in terms of these series is

[tex]\displaystyle\sum_{n=0}^\infty(n+1)a_{n+1}x^n+2\sum_{n=0}^\infty a_nx^n=0[/tex]

[tex]\displaystyle\sum_{n=0}^\infty\bigg(a_{n+1}+2a_n\bigg)x^n=0[/tex]

[tex]\implies\begin{cases}a_0=y(0)\\(n+1)a_{n+1}=-2a_n&\text{for }n\ge0\end{cases}[/tex]

We can solve the recurrence exactly by substitution:

[tex]a_{n+1}=-\dfrac2{n+1}a_n=\dfrac{2^2}{(n+1)n}a_{n-1}=-\dfrac{2^3}{(n+1)n(n-1)}a_{n-2}=\cdots=\dfrac{(-2)^{n+1}}{(n+1)!}a_0[/tex]

[tex]\implies a_n=\dfrac{(-2)^n}{n!}a_0[/tex]

So the ODE has solution

[tex]y(x)=\displaystyle a_0\sum_{n=0}^\infty\frac{(-2x)^n}{n!}[/tex]

which you may recognize as the power series of the exponential function. Then

[tex]\boxed{y(x)=a_0e^{-2x}}[/tex]

Solution of differential equation is, [tex]y=e^{-2x}+c[/tex]

Given differential equation is,

                   [tex]y'+2y=0\\\\\frac{dy}{dx}+2y=0[/tex]

Using separation of variable.

             [tex]\frac{dy}{y}=-2dx[/tex]

Integrating both side.

       [tex]ln(y)=--2x\\\\y=e^{-2x}+c[/tex]

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