5. The differential equation y 00 − xy = 0 is called Airy’s equation, and is used in physics to model the refraction of light. (a) Assume a power series solution, and find the recurrence relation of the coefficients. [Hint: When shifting the indices, one way is to let m = n − 3, then factor out x n+1 and find an+3 in terms of an. Alternatively, you can find an+2 in terms of an−1.] (b) Show that a2 = 0. [Hint: the two series for y 00 and xy don’t “start” at the same power of x, but for any solution, each term must be zero. (Why?)] (c) Find the particular solution when y(0) = 1, y 0 (0) = 0, as well as the particular solution when y(0) = 0, y 0 (0) = 1.

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 Cpk is 0.952

Step-by-step explanation:

The formula to calculate the Cpk of a process is

[tex]Cpk = min(\frac{USL-mean}{3*sigma}, \frac{mean-LSL}{3*sigma} )[/tex]

where

USL (Upper Specification Limit) =3.5cm+0.9cm = 4.4cm

LSL (Lower Specification Limit) =3.5cm-0.9cm=2.6cm

Standard Deviation = sigma = 0.28cm

Mean = 3.4cm

So,

[tex]Cpk=min(\frac{4.4-3.4}{3*0.28} ,\frac{3.4-2.6}{3*0.28})\\\\Cpk=min(\frac{1}{0.84} ,\frac{0.8}{0.84})\\\\Cpk=min(1.190 ,0.952)\\\\\\[/tex]

The Cpk is 0.952

The process capability index or CPk is calculated using the formula min([USL - m]/3σ, [m - LSL]/3σ). In this case, USL is calculated as 4.4 cm and LSL is found to be 2.6 cm. The final CPk value is the smaller of the two resulting values, which in this case is 0.952.

The question asks for the calculation of CPk, which is an index in statistics determining the potential capability of a process in meeting the specification limits. This index considers both the variability of the process and the target in its calculation. The formula for CPk is given by

CPk = min([USL - m]/3σ, [m - LSL]/3σ)

where:

Using the given values from the question,

USL = 3.5 cm + 0.9 cm = 4.4 cm,

LSL = 3.5 cm - 0.9 cm = 2.6 cm,

[USL - m]/3σ = (4.4 cm - 3.4 cm) / (3 * 0.28 cm) = 1.19,

[m - LSL]/3σ = (3.4 cm - 2.6 cm) / (3 * 0.28 cm) = 0.952.

The CPk value will be the smaller of these two values, which is 0.952.

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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:

Given:

Standard deviation or variance of two population.

We need to write a method by which a formal hypothesis test can be conducted of claim made about two population standard deviations or variances.

In General Chi-Square test and F-test are used for variance or standard deviation.

Also Chi-Square test and F-test require that the original population be normally distributed.

Now for Testing a Claim about Variance or Standard Deviation

To test a claim about the value of the variance or the standard deviation of population, then we use the test statistic which follows chi-square distribution with n − 1 degrees of freedom, and is given by the following formula.

[tex]\chi^2=\frac{(n-1)s^2}{\sigma_0^2}[/tex]

Where s is for given standard deviation and [tex][\sigma[/tex] is for claimed standard deviation.

First we make Hypothesis, then we choose the value of α ( level of significance ), after that using above formula we find value of chi-square.

then we find table value for the chosen α also known as table value or p-value. Finally we give final answer by checking relation between p-value and α.

If  p-value < α then null hypothesis is rejected

If p-value > α  then null hypothesis accepted.

there are 52 cards in a deck, 12 of those cards are "face cards", so the remaining are number cards, namely 40.

52 = sample space

40 = favorable outcomes

P(number card | number card) = p(number) * p(number)

so the first time we pull one, there are 52 cards, the probability of a number card is 40/52, or 10/13, and we don't put it back in the deck.

the next time we pull another card, the cards are no longer 52 total, we pulled one out, they're only 51, namely 51 = sample space, and the number cards if we really pulled out before, are no longer 40, are 39, namely 39 = favorable outcomes.

probability of getting a number card the second time?  39/51 or 13/17.

[tex]\bf \stackrel{\textit{probability of getting a number card twice}}{\cfrac{10}{~~\begin{matrix} 13 \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~}\cdot \cfrac{~~\begin{matrix} 13 \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~}{17}\implies \cfrac{10}{17}~~\approx ~~ 0.59}~\hfill 59\%[/tex]

Answer:

105/221

Step-by-step explanation:

There are 52 cards in a deck.

Assuming 2-10 are the number cards

2,3,4,5,6,7,8,9,10 = 9

There are 4 suits

9*4 = 36 cards are number cards

P(1st card is a number card) = number card/ total

                                               =36/52 = 9/13

We do not replace the card, so there are only 51 cards left, and only 35 number cards

P(2nd card is a number card) = number card/ total

                                               =35/51

The probability of getting 2 number cards in a row is

P (number ,number) =P(1st card is a number)*P(2nd card is a number card)  

                                  = 9/13 * 35/51

Dividing the top and bottom by 3      

                                  = 3/13 * 35/17                  

                                  =105/221

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:

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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The answer is:

[tex]t(s(5))=-77[/tex]

To solve the problem, first, we need to compose the functions, and then evaluate the obtained function. Composing function means evaluating a function into another function.

We have that:

[tex]f(g(x))=f(x)\circ g(x)[/tex]

From the statement we know the functions:

[tex]s(x)=-3x-4\\t(x)=4x-1[/tex]

We need to evaluate the function "s" into the function "t", so:

[tex]t(s(x))=4(-3x-4)-1\\\\t(s(x))=-12x-16-1=-12x-17[/tex]

Now, evaluating the function, we have:

[tex]t(s(5))=-12(5)-17=-60-17=-77[/tex]

Have a nice day!

Check the picture below.

Answer:

3x - y = 0; 2x - y = -5

Step-by-step explanation:

Let x be the present age of Hans and y be the present age of Grace,

Since, in present Grace is three times as old as Hans,

⇒ y = 3x

⇒ 3x - y = 0

Now, after 5 years,

The age of Hans = x + 5,

And, the age of Grace = y + 5

Also, in 5 years Grace will be twice as old as Hans is then,

⇒ y + 5 = 2 ( x + 5 )

⇒ y + 5 = 2x + 10

⇒ 2x - y = -5

Hence, the required system of linear equations is,

3x - y = 0; 2x - y = -5

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: [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).

Answer:

225

Step-by-step explanation:

To find your answer, multiply 300 by the decimal form of 75%, which is 0.75.

[tex]300 * 0.75 = 225[/tex]

Answer:

Half of 300 is 150 and half of 150 is 75, which is 25%. 75 x 3 gives us 225. 225 is the answer

Step-by-step explanation:

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

Final answer:

Calculate the probability that the mean life span of a group of 10 hot water heaters is between 12 and 15 years using the standard normal distribution.

Explanation:

The probability that the mean life span in a group of 10 randomly selected hot water heaters is between 12 and 15 years can be calculated using the standard normal distribution.

Given: Mean = 13 years, Standard Deviation = 1.5 years.

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.

Answer:  Probability that either had Internet access or had cable television is 94%.

Step-by-step explanation:

Since we have given that

Probability of households had internet access P(I) = 85% = 0.85

Probability of households had cable television P(C) = 81% = 0.81

Probability of households had both Internet and cable television P(I ∩ C) = 72% = 0.72

We need to find the probability that either had Internet access or had cable television.

As we know the formula:

P(I ∪ C)=P(I) + P(C) - P(I ∩ C)

[tex]P(I\cup C)=0.85+0.81-0.72\\\\P(I\cup C)=0.94=94\%[/tex]

Hence, our required probability is 94%.

To determine the probability of a household having access to either the internet or cable television, the probabilities of both events must be added then subtract the probability of both occurring simultaneously. Doing this calculation, P(Internet) + P(Cable) - P(Both), we find the probability to be 94%.

In order to answer this question, we would need to use some principles of probability, specifically the rule for adding the probabilities of two mutually exclusive outcomes. But, because the outcomes in this case are not mutually exclusive (a household can have both Internet and cable television), we need to adjust our calculation by subtracting the probability of both outcomes occurring together.

The probability of having either Internet or cable television would be P(Internet) + P(Cable) - P(Both) which equals to [tex]85 + 81 - 72 = 94%.[/tex]Therefore, the probability of selecting a household that had either Internet access or cable television is 94%.

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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%.

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:Increasing in x∈(0,π/4)∪(5π/4,2π) decreasing in(π/4,5π/4)

Step-by-step explanation:

given f(x) = sin(x) + cos(x)

f(x) can be rewritten as [tex]\sqrt{2} [\frac{sin(x)}{\sqrt{2} }+\frac{cos(x)}{\sqrt{2} }  ]..................(a)\\\\\ \frac{1}{\sqrt{2} } = cos(45) = sin(45)\\\\[/tex]

Using these result in equation a we get

f(x) = [tex]\sqrt{2} [ cos(45)sin(x)+sin(45)cos(x)]\\\\= \sqrt{2} [sin(45+x)]..........(b)[/tex]

Now we know that for derivative with respect to dependent variable is positive for an increasing function

Differentiating b on both sides with respect to x we get

f '(x) = [tex]f '(x)=\sqrt{2}  \frac{dsin(45+x)}{dx}\\ \\f'(x)=\sqrt{2} cos(45+x)\\\\f'(x)>0=>\sqrt{2} cos(45+x)>0[/tex]

where x∈(0,2π)

we know that cox(x) > 0 for x∈[0,π/2]∪[3π/2,2π]

Thus for cos(π/4+x)>0 we should have

1) π/4 + x < π/2  => x<π/4  => x∈[0,π/4]

2) π/4 + x > 3π/2  => x > 5π/4  => x∈[5π/4,2π]

from conditions 1 and 2 we have  x∈(0,π/4)∪(5π/4,2π)

Thus the function is decreasing in x∈(π/4,5π/4)

To find the intervals of increase and decrease for f(x) = sin x + cos x, we first find the derivative f'(x) = cos x - sin x. By setting the derivative equal to zero, we find the critical points at x = π/4 + kπ. By testing these intervals in the derivative, we can identify the intervals of increase and decrease.

The function given is f(x) = sin x + cos x, which is a combination of a sine and cosine function. To find the intervals where the function is increasing or decreasing, we need to find the derivative of the function first. The derivative of sin x is cos x, and the derivative of cos x is -sin x. So, the derivative of the function f(x) is f'(x) = cos x - sin x.

By setting the derivative equal to zero, cos x - sin x = 0, we can find the critical points where the function may change from increasing to decreasing or vice versa. The solutions to this equation are x = π/4 + kπ, where k is an integer.

Using these points, we can find the intervals of increase and decrease. For instance, if we test a number between 0 and π/4 in the derivative, we find that the function is increasing on the interval (0, π/4). Continuing this process for the rest of the intervals should provide all the intervals of increase and decrease for the function.

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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:

Dimensions of the container should be 12×12×7.5 ft to minimize the making cost.

Step-by-step explanation:

A trash company is designing an open top, rectangular container having volume = 1080 ft³

Let the length of container = x ft , width of the container = y ft and height of the container = z ft.

So volume of the rectangular container = xyz = 1080 ft³

Or [tex]z=\frac{1080}{xy}[/tex] ft -----(1)

Cost of making the bottom of the container = $5 per square ft

Area of the bottom = xy

Cost of making the bottom @ $5 per square ft = 5xy

Area of all sides of the container = 2(xz + yz) = 2z(x+ y)

Now it has been given that cost of making all sides of the container is = $4 per square ft

So total cost to manufacture sides = 4[2z(x + y)]

Now cost of making bottom and sides of the container = 5xy + 8z(x + y)

We put the value of z from equation 1

Total cost A = 5xy+8(x + y)[tex](\frac{1080}{xy})[/tex]

A = 5xy +[tex]8(\frac{1080}{y})+8(\frac{1080}{x})[/tex]

Now we will find the derivative of A and equate it to the zero

[tex]\frac{dA}{dx}=0[/tex] and [tex]\frac{dA}{dy}=0[/tex]

[tex]\frac{dA}{dx}=5y+8(1080)(0)+8(1080)(-\frac{1}{y^{2}})=0[/tex]

5y =[tex]\frac{8\times1080}{y^{2} }[/tex]

5y³ = 8640

y³ =[tex]\frac{8640}{5}=1728[/tex]

y = 12 ft

For [tex]\frac{dA}{dy}=0[/tex]

[tex]\frac{dA}{dy}=5x+\frac{8(-1080)}{x^{2}}[/tex]=0

5x =[tex]\frac{8(1080)}{x^{2} }[/tex]

5x³ = 8640

x³ = 1728

x = 12

Now from equation 1

z =[tex]\frac{1080}{x}[/tex]

  =[tex]\frac{1080}{144}[/tex]

z = 7.5

Therefore, dimensions of the container should be 12×12×7.5 ft to minimize the making cost.

Final answer:

To minimize the total cost, we need to minimize the cost of the bottom and the cost of the sides. We can find the dimensions of the container that will minimize the total cost by solving a system of equations.

Explanation:

Let's assume that the length of the rectangular container is x ft, the width is y ft, and the height is z ft.

The volume of the container is given as 1080 ft3.

Therefore, we have the equation:

x * y * z = 1080

The cost of making the bottom of the container is $5 per square foot and the cost of the sides is $4 per square foot.

The cost of the bottom is 5 * (x * y).

The cost of the sides is 4 * (2xy + 2xz + 2yz).

To minimize the total cost, we need to minimize the cost of the bottom and the cost of the sides.

First, let's solve the volume equation for x:

x = (1080) / (y * z)

Substituting the value of x in the cost equation, we have:

Cost = 5 * (1080) / (y * z) * y + 4 * (2 * (1080 / (y * z)) * y + 2 * (1080) / (y * z) * z + 2 * y * z)

Now, we can find the minimum cost by taking the derivative of the cost equation with respect to y and z, and setting it equal to zero.

Then, we solve the resulting system of equations to find the values of y and z that minimize the cost.

Finally, we substitute the values of y and z back into the volume equation to find the value of x.

By solving the equations, we can find the dimensions of the container that will minimize the total cost.

Answer:

[tex]\frac{1}{y^2}=2x^3+Cx^4[/tex].

Step-by-step explanation:

Given differential equation

[tex]\frac{\mathrm{d}y}{\mathrm{d}x}+\frac{2}{x}y=x^2y^3[/tex]

Differential equation can be write as

[tex]y^{-3}\frac{\mathrm{d}y}{\mathrm{d}x}+\frac{2}{x}y^{-2}=x^2[/tex]

By Bernoulli method

Susbstitute [tex]y^{-2}=t[/tex].....{equationI}

Differentiate equation I w.r.t x then we get

[tex]\frac{\mathrm{d}t}{\mathrm{d}x}=-2y^{-3}\frac{\mathrm{d}y}{\mathrm{d}x}[/tex]

[tex]-\frac{1}{2}\frac{\mathrm{d}t}{\mathrm{d}x}=y^{-3}\frac{\mathrm{d}y}{\mathrm{d}x}[/tex]

Susbstitute the values in the given differential equation then we get

[tex]-\frac{1}{2}\frac{\mathrm{d}t}{\mathrm{d}x}+\frac{2}{x}t=x^2[/tex]

[tex]\frac{\mathrm{d}t}{\mathrm{d}x}-\frac{4}{x}t=-2x^2[/tex]

It is first order linear differential equation and compare with the first order linear differential equation [tex]\frac{\mathrm{d}y}{\mathrm{d}x}+P(x)y=Q(x)[/tex]

Then we get P(x)=[tex]-\frac{4}{x}[/tex] and Q(x)=[tex]-2x^2[/tex]

Integration factor=[tex]e^\intP(x)dx[/tex]

Integration factor= [tex]e^{-\int\frac{4}{x}dx[/tex]

Integration factor= [tex]e^{-4lnx}=e^{lnx^{-4}}=x^{-4}[/tex].

Using [tex]e^{logb}=b[/tex]

[tex]t\times \frac{1}{x^4}=\int{-2x^2}\times\frac{1}{x^4}dx+C[/tex]

[tex]t=-2x^4{\intx^{-2}dx+C}[/tex]

[tex]t=2x^4\times\frac{1}{x}+Cx^4[/tex]

[tex]t=2x^3+Cx^4[/tex]

Substitute [tex]t=\frac{1}{y^2}[/tex] then we get

[tex]\frac{1}{y^2}=2x^3+Cx^4[/tex].

Answer: [tex]\frac{1}{y^2}=2x^3+Cx^4[/tex].

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]

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:

x = 12/7 or 1.7

Step-by-step explanation:

first, cross multiply to get 7x = 12. then, divide 12 by 7 to get 12/7, which can be simplified and rounded to 1.7.

Final answer:

To solve 7/4 = 3/x, use cross multiplication to get 7x = 12, then divide both sides by 7 to find x, which is approximately 1.7 when rounded to the nearest tenth.

Explanation:

To solve the equation 7/4 = 3/x, we can set up a proportion and use cross multiplication. Cross multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction, and setting the products equal to each other. In this case, we multiply 7 by x and 4 by 3 to get the equation 7x = 12.

After cross multiplying, divide both sides of the equation by 7 to solve for x. Doing this, we find that x = 12/7. To convert this to a decimal and round to the nearest tenth, we can divide 12 by 7 using a calculator or long division, resulting in approximately 1.7.

Answer:  1.029 tons/m³

Step-by-step explanation:

Density of sea water may varies for different temperature but at 25°C the density of sea water is taken as 1029 kg/m³.

Density of sea water is 1029 kg/m³

1 tons=1000 kg

1 kg=0.001 tons

Density of sea water=1029×0.001 tons/m³

                                  =1.029 tons/m³

hence the density will be 1.029 tons/m³

Answer:

According to the internet, the density of the water is 1023.6 kg/m^ 3, but when it is converted, it is 1.0273 tons/m^3.

Step-by-step explanation:

This is what I found, hope it helps.

Answer: [tex]H_0:p\leq0.25[/tex]

[tex]H_a:p>0.25[/tex]

Step-by-step explanation:

Given : In the past , 25% of the town residents participated in the school board elections.

Let 'p' be the proportion of voters will participated in the school board elections.

Claim : [tex]p>0.25[/tex]

We know that the null hypothesis has equal sign.

Therefore , the null hypothesis for the given situation will be opposite to the given claim will be :-

[tex]H_0:p\leq0.25[/tex]

And the alternative hypothesis must be :-

[tex]H_a:p>0.25[/tex]

Hence, the correct set of hypotheses is

[tex]H_0:p\leq0.25[/tex]

[tex]H_a:p>0.25[/tex]

Answer:

The lower limits (in 1000 ​cells/mu​L): 0,100, 200, 300, 400, 500, 600 .

The upper limits (in 1000 ​cells/mu​L): 99, 199, 299, 399, 499, 599, 699.

The class width(in 1000 ​cells/mu​L): 100.

Class midpoints (in 1000 ​cells/mu​L): 49.5, 149.5, 249.5, 349.5, 449.5, 549.5, 649.5.

Class boundaries (in 1000 ​cells/mu​L): -0.5, 99.5, 199.5, 299.5, 399.5, 499.5, 599.5, 699.5.

Individuals included in the summary: 155.

Step-by-step explanation:

For the lower class limit is needed the smallest value in each class: 0,100, 200, 300, 400, 500, 600.

For the upper-class limit is needed the biggest value in each class: 99, 199, 299, 399, 499, 599, 699.

The class width is the difference between the lower limit of one class and the lower limit of the previous class. For example, 200 is the lower limit of one class and the lower limit of the previous class is 100, so 200-100=100.

Class midpoints are the average of the limits of a class if the limits are 0 and 99 then:

[tex]Midpoint=\frac{0+99}{2}=49.5[/tex]

[tex]Midpoint=\frac{100+199}{2}=149.5[/tex]

[tex]Midpoint=\frac{200+299}{2}=249.5[/tex]

[tex]Midpoint=\frac{300+399}{2}=349.5[/tex]

[tex]Midpoint=\frac{400+499}{2}=449.5[/tex]

[tex]Midpoint=\frac{500+599}{2}=549.5[/tex]

[tex]Midpoint=\frac{600+699}{2}=649.5[/tex]

Class boundaries are the numbers than allow to separate each class, to find each one, first calculate the gap between each class (the lower limit of one class minus the upper limit of the previous one) and divide by 2:

100-99=1

1/2=0.5

Add this number to all the upper limit to find the upper boundaries:

[tex]99+0.5=99.5[/tex]

[tex]199+0.5=199.5[/tex]

[tex]299+0.5=299.5[/tex]

[tex]399+0.5=399.5[/tex]

[tex]499+0.5=499.5[/tex]

[tex]599+0.5=599.5[/tex]

[tex]699+0.5=699.5[/tex]

And  subtract this number (0.5) from the lower limit to find the lower boundaries:

[tex]0-0.5=-0.5[/tex]

[tex]100-0.5=99.5[/tex]

[tex]200-0.5=199.5[/tex]

[tex]300-0.5=299.5[/tex]

[tex]400-0.5=399.5[/tex]

[tex]500-0.5=499.5[/tex]

[tex]600-0.5=599.5[/tex]

The total of individuals is equal to the sum of all the frequencies of each class:

[tex]3+54+76+21+0+0+1= 155[/tex]

The lower class limits are 0, 100, 200, etc. The class width is 100. The summation of the frequencies, which equals the total number of individuals, is 155.

In the given data, the lower class limits are the smallest numbers in each class or group, which are 0, 100, 200, 300, 400, 500, and 600 (in 1000 cells/muL). The upper class limits, which are the highest numbers in each class, are 99, 199, 299, 399, 499, 599, and 699.

The class width, which is the difference between two consecutive lower class limits, is 100. The class midpoints can be calculated by adding the lower and upper limits of each class and divide by 2, yielding 49.5, 149.5, 249.5, etc. The class boundaries are the numbers that separate classes without leaving gaps: -0.5, 99.5, 199.5, etc.

The total number of individuals included in the summary is the sum of the frequencies, which equals 155 in this case.

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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.