Is 4320 perfect, abundant, or deficient? also perfect numbers? Explain why all positive multiples of 6 greater than 6 are aburndant numbers.

Answer:

Angle = 35°

Step-by-step explanation:

The figure would form a triangle and the tree will divide it into two 90 degree triangles.

Height of the triangles will be 8 feet which is opposite to the angle you have to find.

Hypotenuse of both 90 degree triangles will be 14 feet.

To find the angle, use the sin formula

sin (angle) = opposite/hypotenuse

angle = ?

opposite = 8

hypotenuse = 14

sin (angle) = 8/14

angle = sin inverse (0.57)

angle = 34.8° rounded off to 35°

The angle will be the same for both triangles.

Therefore, the acute angle each triangle makes with the ground is 35°.

Answer:

  both domain and range are {-4, -3, -2, -1, 0, 1, 2, 3, 4}

  it IS a function

Step-by-step explanation:

The domain is the list of x-values of the plotted points. It is all the integers from -4 to +4, inclusive.

The range is the list of y-values of the plotted points. It is all the integers from -4 to +4, inclusive. (domain and range are the same for this function)

No x-value has more than one y-value associated with it, so this relation IS A FUNCTION.

Answer:

Female

Step-by-step explanation:

The formula for finding z-score is;

z=(x-μ)/δ

where x= weight given=1600g

μ=mean weight given

δ=standard deviation given

For male

x=1600g

μ=3232.9g

δ=714.6g

z=(1600-3232.9)/714.6

z= -2.3

The weight for the male is 2.3 standard deviations below the mean

For female

x=1600g

μ=3094.9g

δ=586.3g

z=(1600-3094.9)/586.3

z= -2.5

The weight for the female is 2.5 standard deviations below the mean

⇒The female has the weight that is more extreme

Answer:

z= 1

Step-by-step explanation:

got it on egde

Answer:

1) 0.6% of the production is defective.

2) Ana will owe $570.

Step-by-step explanation:

Both questions here are percentage problems

Percentage problems can be explained as a rule of three problem

In a rule of three problem, the first step is identifying the measures and how they are related, if their relationship is direct of inverse.

When the relationship between the measures is direct, as the value of one measure increases, the value of the other measure is going to increase too. In this case, the rule of three is a cross multiplication.

When the relationship between the measures is inverse, as the value of one measure increases, the value of the other measure will decrease. In this case, the rule of three is a line multiplication.

Percentage problems have a direct relationship between the measures.

1) The problem states that a quality control process finds 46.8 defects for every 7,800 units of production. And asks what percent of the production is defective? We have to answers how many defects are there for 100 units of production. So:

46.8 defects - 7,800 units

x defects - 100 units

7,800x = 4680

[tex]x = \frac{4680}{7800}[/tex]

x = 0.6

0.6% of the production is defective.

2) The problem states that the medical insurance policy requires Ana to pay the first $100 of her hospital expense. The insurance company will then pay 90% of the remaining expense. The total bill is expected to be about $4,800.

Ana has to pay:

P = P1 + P2

-P1 :The first $100

-P2: 10% of the remaining expense. The remaining expense is $4,800-$100 = $4,700. Ana has to pay 10% of this. So

4700 - 100%

P2 - 10%

100P2 = 47000

[tex]P2 = \frac{47000}{100}[/tex]

P2 = $470

Ana will owe P = P1 + P2 = $100 + $470 = $570.

To find the percentage of defective units in the production, set up a ratio and calculate the percentage. Approximate 0.6% of the production is defective. To find how much Ana will owe of the total bill, subtract $100 from the total bill and calculate 10% of the remaining expense. Ana will owe $470 of the total bill.

To find the percent of the production that is defective, we need to find the ratio of the number of defective units to the total number of units produced. In this case, we have 46.8 defects for every 7,800 units of production. So, we can set up the ratio as:

Defective Units / Total Units = 46.8 / 7,800

To find the percentage, we can multiply the ratio by 100:

Percentage of Defective Units = (46.8 / 7,800) * 100

Now, calculate the value of the ratio and simplify to find the percentage:

Percentage of Defective Units = 0.006 * 100 = 0.6%

Therefore, approximately 0.6% of the production is defective.

For the second question, to find how much of the total bill Ana will owe, we need to calculate the 10% of the remaining expense after she pays the first $100. First, subtract $100 from the total bill:

Remaining Expense = $4,800 - $100 = $4,700

Next, calculate 10% of the remaining expense:

Amount Ana will owe = 10% of $4,700 = ($4,700 * 10) / 100 = $470

Therefore, Ana will owe $470 of the total bill.

Answer:

a) The patient is going to need 3.2 pills. So four doses.

b) The last dose is at midnight, that i consider 12 PM

Step-by-step explanation:

This problem can be solved by a rule of three in which the measures are directly related, meaning that we have a cross multiplication.

a) How many pills will this patient need?

The first step is finding how many mg she is going to need.

She weighs 50kg, and the total daily amount cannot exceed 8 mg per kg of body. So:

1kg - 8 mg

50 kg - x mg

[tex]x = 50*8[/tex]

[tex]x = 400[/tex]mg.

Each dose has 125 mg, so she will need:

1 dose - 125 mg

x doses - 400mg

[tex]125x = 400[/tex]

[tex]x = \frac{400}{125}[/tex]

[tex]x = 3.2[/tex]

The patient is going to need 3.2 pills. So four doses.

b) If the first dose is administered at 6 AM, when is the last dose given?

There are four doses, administered every 6 hours.

6AM: First dose

12AM: Second dose

6PM: Third dose

12PM: Fourth Dose

The last dose is at midnight, that i consider 12 PM

Final answer:

To calculate the number of pills, we first determine the total daily allowable dosage based on the patient's weight and divide that by the dosage per pill, rounding up to provide whole pills. Then we schedule the medication every 6 hours starting from 6 AM, with the last dose at 12 AM.

Explanation:

The question involves calculating the appropriate dosage of medication for a patient based on weight and the maximum allowable daily dosage.

The medication prescribed is available as enteric-coated pills of 125mg each, and the patient requires a dosage that does not exceed 8mg per kg of their body weight per day.

The patient weighs 50 kg. To find the total daily dosage allowed, we multiply the patient's weight by the maximum mg per kg: 50 kg × 8 mg/kg = 400 mg.

Since the patient can have 400 mg of medication per day and the medication comes in 125 mg pills, we divide the total daily dosage by the dosage per pill: 400 mg ÷ 125 mg/pill = 3.2 pills.

Since we cannot give a patient a fraction of a pill, the patient will need 4 pills per day.

The medication must be administered every 6 hours. If the first dose is at 6 AM, the subsequent doses will be at 12 PM, 6 PM, and 12 AM, with the last dose given precisely 18 hours after the first dose.

Answer: Algorithm complexity or the complexity of an algorithm is known as a measure under which one evaluates degree of count of operations, that are specifically performed by an algorithm which is taken in consideration as a function of size of the data. In rudimentary terms, it is referred to as a rough approximation of number of stages required in order to enforce an algorithm.

Answer:

The average cost per item when 200 items are produced is 61.6

Step-by-step explanation:

We start with the cost formula given by:

[tex]C(x)=7.6x+10,800[/tex]

Then we compute C(x) for x=200, 2000 and 5000 as follows:

[tex]C(200)=7.6*200+10,800=12,320\\C(2000)=7.6*2000+10,800=26,000\\C(5000)=7.6*5000+10,800=48,800[/tex]

Finally, to obtain the average cost per item when 200, 2,000 and 5,000 are produced (we will denote this by Av(200), Av(2000) and Av(5000) respectively) we just need to divide C(x) by the number of items produced. Then [tex]Av(x)=\frac{C(x)}{x}[/tex].

[tex]Av(200)=\frac{C(200)}{200}=\frac{12,320}{200}= 61.6\\Av(2000)=\frac{C(2000)}{2000}=\frac{26,000}{2000}= 13\\Av(5000)=\frac{C(5000)}{5000}=\frac{48,800}{5000}= 9.76\\[/tex]

Answer:

Marginal revenue = R'(Q) = -0.6 Q + 221

Average revenue = -0.3 Q + 221

Step-by-step explanation:

As per the question,

Functions associated with the demand function P= -0.3 Q + 221, where Q is the demand.

Now,

As we know that the,

Marginal revenue is the derivative of the revenue function, R(x), which is equals the number of items sold,

Therefore,

R(Q) = Q × ( -0.3Q + 221) = -0.3 Q² + 221 Q

∴ Marginal revenue = R'(Q) = -0.6 Q + 221

Now,

Average revenue (AR) is defined as the ratio of the total revenue by the number of units sold that is revenue per unit of output sold.

[tex]Average\ revenue\ = \frac{Total\ revenue}{number\ of\ units\ sold}[/tex]

Where Total Revenue (TR) equals quantity of output multiplied by price per unit.

TR = Price (P) × Total output (Q) = (-0.3Q + 221) × Q = -0.3 Q² + 221 Q

[tex]Average\ revenue\ = \frac{TR}{Q}[/tex]

[tex]Average\ revenue\ = \frac{-0.3Q^{2}+221Q}{Q}[/tex]

∴ Average revenue = -0.3Q + 221

Answer:

40 bottles

Step-by-step explanation:

In the problem it is said that there were sold eigth times as much Cabernet Sauvignon as Zinfandel, this menas that if for example the cafe sold 1 bottle of zinfandel it would be 8 bottles of Cabernet, so we have to multiply the number of bottles sold fon Zinfandel by 8, as there were 5 bottles of sauvingon sold we have to multiply

8x5=40 bottles.

The Cabernet Cafe sold 40 bottles of Cabernet Sauvignon, given that it sold 8 times as many as the 5 bottles of Zinfandel.

The question involves a simple proportion calculation, typical of mathematics problems. Given the information that the Cabernet Cafe sold eight times as much Cabernet Sauvignon as Zinfandel, we can set up a ratio and solve for the unknown. In this case, we know that the number of Cabernet Sauvignon sold is 8 times the amount of Zinfandel sold. If the cafe sold 5 bottles of Zinfandel, using the said proportion, we multiply 5 (number of Zinfandel bottles sold) by 8 (the given ratio) to find the number of Cabernet Sauvignon bottles sold - which is 40 bottles.

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Answer:

Water should be greater than 946.11 gallons per week to prevent the pool from drying.

Step-by-step explanation:

You have an outdoor swimming pool that is 5.0 m wide and 12.0 m long.

The height of water evaporated is = 2.35 inch

We will convert this to m.

1 inch = 0.0254 meter

So, 2.35 inch = [tex]2.35\times0.0254=0.05969[/tex] meter

Now, volume of water in the pool = [tex]5\times12\times0.05969=3.5814[/tex] cubic meter per week.

1 cubic meter = 264.172 gallons

So, 3.5814 cubic meter = [tex]3.5814\times264.172=946.11[/tex]gallons

Hence, the volume of the water that should be poured in the swimming pool should be greater than 946.11 gallons per week to prevent the pool from drying.

Answer:

b. 5  

Step-by-step explanation:

In a stem and leaf plot,  each observation is split into a stem (the first digit or digits) and a leaf (usually the last digit).  

In this example, the entries in the 20s, 30s, and 40s are:

20, 20, 24, 25, 26, 35, 36, 43, 43

We omit 20, 20, 43, and 43 because they are not between 20 and 40.

That leaves the five observations

24, 25, 26, 35, 36

Answer:

less than

Step-by-step explanation:

When we are performing a two-tailed test of whether μ = 100,  the probability of detecting a shift of the mean to 105 will be ___Less__than___ the probability of detecting a shift of the mean to 110.

this means that

mean of 105 < mean of 110.

A volt in terms of meters, seconds, kilograms, and coulombs is

[tex]$\frac{\text{kilogram} \times \text{meter}^2}{\text{second}^2 \times \text{coulomb}}$[/tex]

We are given that;

(vf)α=−2qαΔVmα

Now,

To find the equivalent of a volt in terms of basic SI units, we can use the definition of a volt as the potential difference that causes one joule of energy to be transferred per coulomb of charge.

A joule is the unit of energy, which is defined as the work done by a force of one newton over a distance of one meter.

A newton is the unit of force, which is defined as the product of mass and acceleration. Therefore, we can write:

[tex]$1 \text{ volt} = \frac{1 \text{ joule}}{1 \text{ coulomb}} = \frac{1 \text{ newton} \times 1 \text{ meter}}{1 \text{ coulomb}} = \frac{1 \text{ kilogram} \times 1 \text{ meter} \times 1 \text{ meter}}{1 \text{ second}^2 \times 1 \text{ coulomb}}$[/tex]

Simplifying, we get:

[tex]$1 \text{ volt} = \frac{\text{kilogram} \times \text{meter}^2}{\text{second}^2 \times \text{coulomb}}$[/tex]

Therefore, by volt answer will be [tex]$\frac{\text{kilogram} \times \text{meter}^2}{\text{second}^2 \times \text{coulomb}}$[/tex].

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A volt is a unit of potential difference, which is measured in joules per coulomb (V). 1 volt is equal to 1 joule per coulomb (1 V = 1 J/C).

The potential difference between two points A and B, VB - VA, is defined as the change in potential energy divided by the charge q. This potential difference is measured in joules per coulomb, which is called a volt (V). To express a volt in terms of the basic SI units of meters (m), seconds (s), kilograms (kg), and coulombs (C), we need to use the equation J = V × C, where J represents the unit of energy, the joule.

We can rewrite the equation J = V × C as V = J / C. Since 1 V is equivalent to 1 J/C, we can express the volt in terms of meters (m), seconds (s), kilograms (kg), and coulombs (C) as:

1 V = 1 J/C

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Answer:

There is no solution for this system

Step-by-step explanation:

I am going to solve this system by the Gauss-Jordan elimination method.

The Gauss-Jordan elimination method is done by transforming the system's augmented matrix into reduced row-echelon form by means of row operations.

We have the following system:

[tex]2x_{1} - x_{2} + 3x_{3} = -10[/tex]

[tex]x_{1} - 2x_{2} + x_{3} = -3[/tex]

[tex]x_{1} - 5x_{2} + 2x_{3} = -7[/tex]

This system has the following augmented matrix:

[tex]\left[\begin{array}{ccc}1&-3&4|-4\\3&-7&7|-8\\-4&6&-1|7\end{array}\right][/tex]

We start reducing the first row. So:

[tex]L2 = L2 - 3L1[/tex]

[tex]L3 = L3 + 4L1[/tex]

Now the matrix is:

[tex]\left[\begin{array}{ccc}1&-3&4|-4\\0&2&-5|4\\0&-6&15|-9\end{array}\right][/tex]

We divide the second line by 2:

[tex]L2 = \frac{L2}{2}[/tex]

And we have the following matrix:

[tex]\left[\begin{array}{ccc}1&-3&4|-4\\0&1&\frac{-5}{2}|2\\0&-6&15|-9\end{array}\right][/tex]

Now we do:

[tex]L3 = L3 + 6L2[/tex]

So we have

[tex]\left[\begin{array}{ccc}1&-3&4|-4\\0&1&\frac{-5}{2}|2\\0&0&0|3\end{array}\right][/tex]

This reduced matrix means that we have:

[tex]0x_{3} = 3[/tex]

Which is not possible

There is no solution for this system

Answer:

  28.4 units

Step-by-step explanation:

If we call the given angles C and A, then the given side is c, and the other two sides can be found from the Law of Sines.

Angle B is the remaining angle of the triangle:

  180° -C -A = 180° -30° -45° = 105°

The remaining sides are ...

  b = sin(B)/sin(C)·c = sin(105°)/sin(30°)·6√2 ≈ 16.4

  a = sin(A)/sin(C)·c = sin(45°)/sin(30°)·6√2 = 12

Then the sum of the lengths of the remaining sides is ...

  a + b = 12 + 16.4 = 28.4 . . . units

By applying the principles of trigonometry and the Pythagorean theorem, we can determine that the sum of the lengths of the two remaining sides is approximately 25.5 units.

This question is about the application of trigonometric principles and the Pythagorean theorem. In a triangle, the sine of an angle is defined as the ratio of the side opposite that angle to the hypotenuse. Given a 30-degree angle and its opposite side of length 6√2, we can find the hypotenuse (h) using the fact that sin(30) = 1/2. So, 1/2 = 6√2 / h. Solving this, we get h = 2 * 6√2 = 12√2.

The angle 45 degrees helps determine the length of the remaining side. Using the fact that cos(45) = s/h, where s is the remaining side, we get cos(45) = s / 12√2. Solving for s, s = cos(45) * 12√2, s = √2/2 * 12√2 = 6√2.

So, the sum of the lengths of the two remaining sides is 12√2 + 6√2 = 18√2, which is approximately 25.5 when rounded to the nearest tenth.

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Answer:

x = 10

Step-by-step explanation:

8 + 12 = 2x

8 + 12 = 20

20 = 2x

----   -----

2      2

10 = x

x = 10

Hey!

----------------------------------------------------

Solution:

8 + 12 = 2x

~Divide 2 to both sides

20/2 = 2x/2

~Simplify

10 = x

----------------------------------------------------

Answer:

x = 10

----------------------------------------------------

Hope This Helped! Good Luck!

Answer:

214

Step-by-step explanation:

The playing field is 53.33 yards wide and 120 yards long you would need to find the area so multiply 53.33 by 120 yards. That equals 6399.6 , the thickness they are applying is 1.2 millimeters. You would divide the area, 6399.6 by 1.2 which would equal 5333. Divide that by 25 gallons and it equals 213.32, you would need to purchase 214, rounded.

Answer:

The number of containers to purchase is   [tex]N_V= 67.85[/tex]

Step-by-step explanation:

From the question we are told that

        The playing field  width is [tex]w_f = 53.33 \ yard = 53.33*0.9144 = 48.76m[/tex]

        The playing field length is [tex]l_f = 120 \ yards = 120 * 0.9144 = 109.728m[/tex]

The volume of one container is [tex]V= 25 \ gallon = 25 * 0.00378541 = 0.094625m^3[/tex]

        The thickness of the painting is  [tex]t = 1.2 \ mm = 1.2 * 0.001 = 0.0012m[/tex]

The area of the playing field is [tex]A = 48.76 * 109.728[/tex]

                                         [tex]=5350.337m^2[/tex]

The number of container of paint needed [tex]N_V[/tex] [tex]= \frac{area \ of \ playing \ field(A) * thickness \ of \ paint \ application(t) }{volume\ single \ container(V)}[/tex]

=>    [tex]N_V = \frac{5350.337 * 0.0012}{0.094625}[/tex]

             [tex]N_V= 67.85[/tex]

We are 80% confident that the average net change in a student's score after completing the course lies between 7.88 and 21.12 points.

Here's how to construct the 80% confidence interval:

1. Calculate the sample mean and standard deviation:

Sample mean (xbar) = (23 + 18 + 23 + 12 + 13 + 23) / 6 = 18.33 points

Sample standard deviation (s) = √[(23 - 18.33)^2 + (18 - 18.33)^2 + ... + (23 - 18.33)^2] / (6 - 1) ≈ 5.38 points

2. Find the critical value for a 80% confidence interval:

For a two-tailed 80% confidence interval with 5 degrees of freedom (n-1), the critical value from the t-distribution table is approximately 1.942.

3. Calculate the margin of error:

Margin of error (ME) = critical value * standard deviation / √n

ME = 1.942 * 5.38 / √6 ≈ 4.49 points

4. Construct the confidence interval:

Lower limit = sample mean - margin of error = 18.33 - 4.49 ≈ 13.84 points

Upper limit = sample mean + margin of error = 18.33 + 4.49 ≈ 22.82 points

Therefore, we can be 80% confident that the true average net change in a student's score after completing the course falls within the range of 13.84 to 22.82 points, or rounded to the nearest whole number, 7.88 to 21.12 points.

Note: This calculation assumes the population is approximately normal. If there is reason to believe the population is not normal, a different method like bootstrapping may be more appropriate.

Answer:

There is a 0.64% probability that the costumer has filed a claim.

Step-by-step explanation:

Probability:

What you want to happen is the desired outcome.

Everything that can happen iis the total outcomes.

The probability is the division of the number of possible outcomes by the number of total outcomes.

Our problem has these following probabilities:

-78% that a costumer is a good risk.

-20% that a costumer is a medium risk.

-2% that a costumer is a poor risk.

Also:

- 0.5% of a good risk costumer filling an accident claim

- 1% of a medium risk costumer filling an accident claim.

-2.5% of a poor risk costumer filling an accident claim.

The question is:

What is the probability that the customer has filed a claim?

[tex]P = P_[1} + P_{2} + P_{3}[/tex], in which:

-[tex]P_{1}[/tex] is the probability that a good risk costumer is chosen and files a claim. This probability is: the probability of a good risk costumer being chosen multiplied by the probability that a good risk costumer files a claim. So:

[tex]P_[1} = 0.78*0.005 = 0.0039[/tex]

-[tex]P_{2}[/tex] is the probability that a medium risk costumer is chosen and files a claim. This probability is: the probability of a medium risk costumer being chosen multiplied by the probability that a medium risk costumer files a claim. So:

[tex]P_[2} = 0.20*0.01 = 0.002[/tex]

-[tex]P_{3}[/tex] is the probability that a poor risk costumer is chosen and files a claim. This probability is: the probability of a poor risk costumer being chosen multiplied by the probability that a poor risk costumer files a claim. So:

[tex]P_[3} = 0.02*0.025 = 0.0005[/tex]

[tex]P = P_[1} + P_{2} + P_{3} = 0.0039 + 0.002 + 0.0005 = 0.0064[/tex]

There is a 0.64% probability that the costumer has filed a claim.

The probability that a randomly selected customer has filed a claim is calculated using the total law of probability, which yields a result of 0.0064 or 0.64%, after considering the probabilities of each risk group filing a claim.

To calculate the probability that a randomly chosen customer has filed a claim, we need to use the total law of probability. This involves multiplying the probability of a customer being in each risk group by the probability that a customer in that risk group files a claim, and then summing these products.

For good risks, this probability is 0.78 (the percentage of good risk customers) multiplied by 0.005 (the probability a good risk customer files a claim): 0.78 * 0.005 = 0.0039.

For medium risks, the calculation is 0.20 * 0.01 = 0.0020.

For poor risks, the calculation is 0.02 * 0.025 = 0.0005.

Adding these probabilities together gives us the total probability that a customer has filed a claim: 0.0039 + 0.0020 + 0.0005 = 0.0064, or 0.64% if expressed as a percentage.

Therefore, the probability that a randomly selected customer from the insurance company has filed a claim is 0.0064 or 0.64%, rounded to four decimal places.

Answer:  a) 0.40    b) 0.15

Step-by-step explanation:

Let A denotes the event that students wear a watch and B denotes the event that students wear a bracelet.

Given : P(A)=0.25   ;   P(B)=0.30

[tex]P(A'\cup B')=0.60[/tex]

Since, [tex]P(A\cup B)=1-P(A'\cup B')=1-0.60=0.40[/tex]

Thus, the probability that this student is wearing a watch or a bracelet = 0.40

Also, [tex]P(A\cup B)=P(A)+P(B)-P(A\cap B)[/tex]

[tex]P(A\cap B)=P(A)+P(B)-P(A\cup B)[/tex]

[tex]P(A\cap B)=0.25+0.30-0.40\\\\\Rightarrow\ P(A\cap B)=0.15[/tex]

Thus,  the probability that this student is wearing both a watch and a bracelet= 0.15

Answer:

Step-by-step explanation:

Given that at a certain school, twenty-five percent of the students wear a watch and thirty percent wear a bracelet.

A- people who wear watch = 25%

B - people who wear bracelet = 30%

(AUB)' - People who wear neither a watch nor a bracelet=60%

[tex]A \bigcap B[/tex] - People who wear both =100%-60% = 40%

a) [tex]P(AUB) = P(A)+P(B)-P(AB) = 25%+30%-40%\\= 15%[/tex]

b) the probability that this student is wearing both a watch and a bracelet

= [tex]P(A \bigcap B) = 40%[/tex]

Answer:

Everything is verified in the step-by-step explanation.

Step-by-step explanation:

We have the following differential equation:

[tex]x'' + 2x' + x = 0[/tex]

This differential equation has the following characteristic polynomial:

[tex]r^{2} + 2r + 1 = 0[/tex]

This polynomial has two repeated roots of [tex]r = -1[/tex].

Since the roots are repeated, our solution has the following format:

[tex]x(t) = c_{1}e^{-t} + c_{2}te^{-t}[/tex]

This shows that the family of solutions satisfies the equation for all values of the constants. The values of the constants depends on the initial conditions.

Lets solve the system with the initial conditions given in the exercise.

[tex]x(0) = 8[/tex]

[tex]c_{1}e^{0} + c_{2}(0)e^{0} = 8[/tex]

[tex]c_{1} = 8[/tex]

--------------------

[tex]x'(0) = 8[/tex]

[tex]x(t) = c_{1}e^{-t} + c_{2}te^{-t}[/tex]

[tex]x'(t) = -c_{1}e^{-t} + c_{2}e^{-t} - c_{2}te^{-t}[/tex]

[tex]-c_{1}e^{0} + c_{2}e^{0} - c_{2}(0)e^{0} = 8[/tex]

[tex]-c_{1} + c_{2} = 8[/tex]

[tex]c_{2} = 8 + c_{1}[/tex]

[tex]c_{2} = 8 + 8[/tex]

[tex]c_{2} = 16[/tex]

With these initial conditions, we have the following solution

[tex]x(t) = 8e^{-t} + 16te^{-t}[/tex]

Answer with Step-by-step explanation:

We are given that A and B are two countable sets

We have to show that if A and B are countable then [tex]A\cup B[/tex] is countable.

Countable means finite set or countably infinite.

Case 1: If A and B are two finite sets

Suppose A={1} and B={2}

[tex]A\cup B[/tex]={1,2}=Finite=Countable

Hence, [tex]A\cup B[/tex] is countable.

Case 2: If A finite and B is countably infinite

Suppose, A={1,2,3}

B=N={1,2,3,...}

Then, [tex]A\cup B[/tex]={1,2,3,....}=N

Hence,[tex]A\cup B[/tex] is countable.

Case 3:If A is countably infinite and B is finite set.

Suppose , A=Z={..,-2,-1,0,1,2,....}

B={-2,-3}

[tex]A\cup B[/tex]=Z=Countable

Hence, [tex]A\cup B[/tex] countable.

Case 4:If A and B are both countably infinite sets.

Suppose A=N and B=Z

Then,[tex]A\cup B[/tex]=[tex]N\cup Z[/tex]=Z

Hence,[tex]A\cup B[/tex] is countable.

Therefore, if A and B are countable sets, then [tex]A\cup B[/tex] is also countable.

Answer:

To remedy confusions like yours and to avoid the needless case analyses, I prefer to define X to be countable if there is a surjection from N to X.

This definition is equivalent to a few of the many definitions of countability, so we are not losing any generality.

It is a matter of convention whether we allow finite sets to be countable or not (though, amusingly, finite sets are the only ones whose elements you could ever finish counting).

So, if A and B be countable, let f:N→A and g:N→B be surjections. Then the two sequences (f(n):n⩾1)=(f(1),f(2),f(3),…) and (g(n):n⩾1)=(g(1),g(2),g(3),…) eventually cover all of A and B, respectively; we can interleave them to create a sequence that will surely cover A∪B:

(h(n):n⩾1):=(f(1),g(1),f(2),g(2),f(3),g(3),…).

An explicit formula for h is h(n)=f((n+1)/2) if n is odd, and h(n)=g(n/2) if n is even.

Hope it helps uh mate...✌

Answer:

Kay's husband drove at a speed of 50 mph

Step-by-step explanation:

This is a problem of simple motion.

First of all we must calculate how far Kay traveled to her job, and then estimate the speed with which her husband traveled later.

d=vt

v=45 mph

t= 20 minutes/60 min/hour = 0.333 h (to be consistent with the units)

d= 45mph*0.333h= 15 miles

If Kay took 20 minutes to get to work and her husband left home two minutes after her and they both arrived at the same time, it means he took 18 minutes to travel the same distance.

To calculate the speed with which Kate's husband made the tour, we will use the same initial formula and isolate the value of "V"

d=vt; so

v=[tex]\frac{d}{t}[/tex]

d= 15 miles

t= 18 minutes/60 min/hour = 0.30 h  (to be consistent with the units)

v=[tex]\frac{d}{t}=\frac{15 miles}{0.3 h}=50mph[/tex]

Kay's husband drove at a speed of 50 mph

A.

Let Annie's weight be = a

Let Benjie's weighs = b

Let Carmen's weight be = c

One day Annie weighed 24 ounces more than Benjie, equation forms:

[tex]a=b+24[/tex]        ......(1)

Benjie weighed 3 1/4 pounds less than Carmen.  

In ounces:

1 pound = 16 ounces

[tex]\frac{13}{4}[/tex] pounds = [tex]\frac{13}{4}\times16=52[/tex] ounces

[tex]b=c-52[/tex]  or

[tex]c=b+52[/tex]    ......(2)

Now adding (1) and (2), we get

a+b=b+24+c-52

=> [tex]a=c-28[/tex]

This gives  Annie weighs 28 ounces less than Carmen.

B.

We cannot know anyone's actual weight, as we only know their relative weights.

Answer: 532

Step-by-step explanation:

Let x be the number of students and y be the number of non-students.

Then,  by considering the given information, we have

[tex]x+y=817-----(1)\\\\2x+3y=1919-----------(2)[/tex]

Multiply 2 on the both sides of equation (1), we get

[tex]2x+2y=1634--------(3)[/tex]

Subtract (3) from (2), we get

[tex]y=285[/tex]

Put the value of y in (1), we get

[tex]x+285=817\\\\\Rightarrow\ x=817-285=532[/tex]

Hence, 532 student tickets were sold .

Answer:

In the first picture, correct answer is b: the inverse of function q is function r, because they are symmetrical about the line y = x.

In the second picture, correct answer is b: {y| 0<= y < 8}.

Step-by-step explanation:

The inverse function of f ( f⁻¹(x) ) must satisfy that: f(f⁻¹(x)) = f⁻¹(f(x)) = x; it returns every point x, transformed under function f, to its original place. In the graph, this property translates in the following statement: the inverse function of a function f is the reflection over the identity function (y = x).

In the graph shown in the question, the blue graph (r), is the one that corresponds to the reflection under the identity function. Therefore, the correct answer is b.

Regarding the second picture, first we need to understand what the range means. The range of a function corresponds to the set of all resulting values of the dependent variable. Since the values taken by the dependent variable span from 0 to 8 (including the 0 but not including the 8), then the answer is b: {y| 0<= y < 8}.

[tex]\left[\begin{array}{ccc}a_{1}&b_{1}&c_{1}\\a_{2}&b_{2}&c_{2}\\a_{3}&b_{3}&c_{3}\end{array}\right]=\left[\begin{array}{ccc}-a_{1}^{2}-b_{1}^{2}\\-a_{2}^{2}-b_{2}^{2}\\-a_{3}^{2}-b_{3}^{2}\end{array}\right][/tex]

In the question,

We have to find out the circumcentre of the circle passing through the triangle with the vertices (a₁, b₁), (a₂, b₂) and (a₃, c₃).

So,

The circle is passing through these points the equation of the circle is given by,

[tex]x^{2}+y^{2}+ax+by+c=0[/tex]

On putting the points in the circle we get,

[tex]x^{2}+y^{2}+ax+by+c=0\\(a_{1})^{2}+(b_{1})^{2}+a(a_{1})+b(b_{1})+c=0\\and,\\(a_{2})^{2}+(b_{2})^{2}+a(a_{2})+b(b_{2})+c=0\\and,\\(a_{3})^{3}+(b_{3})^{3}+a(a_{3})+b(b_{3})+c=0\\[/tex]

So,

[tex](a_{1})^{2}+(b_{1})^{2}+a(a_{1})+b(b_{1})+c=0\\a(a_{1})+b(b_{1})+c=-(a_{1})^{2}-(b_{1})^{2}\,.........(1)\\and,\\a(a_{2})+b(b_{2})+c=-(a_{2})^{2}-(b_{2})^{2}\,.........(2)\\and,\\a(a_{3})+b(b_{3})+c=-(a_{3})^{3}-(b_{3})^{3}\,.........(3)\\[/tex]

On solving these equation using, Matrix method we can get the required equation of the circle,

[tex]\left[\begin{array}{ccc}a_{1}&b_{1}&c_{1}\\a_{2}&b_{2}&c_{2}\\a_{3}&b_{3}&c_{3}\end{array}\right]=\left[\begin{array}{ccc}-a_{1}^{2}-b_{1}^{2}\\-a_{2}^{2}-b_{2}^{2}\\-a_{3}^{2}-b_{3}^{2}\end{array}\right][/tex]

This is the required answer.

The center of the circumscribed circle of the symmetric isosceles triangle is at the origin, and the radius is equal to the length of the triangle's equal sides, denoted as r.

To find the center and the radius of the circle which circumscribes the triangle with vertices at ai, a, and a3, we must first understand the nature of the triangle. Given that the triangle is described to be symmetric with equal sides AB = BC = r, it is an isosceles triangle. The perpendicular bisector of the base a will pass through the midpoint of the base and the opposite vertex, given it is symmetric about this bisector. This will also be the diameter of the circumscribed circle. Consequently, as the triangle is isosceles and symmetric, we can use the properties of similar isosceles triangles to solve for the center and radius of the circumscribed circle.

Since the base a will also be the diameter of the circumscribed circle, and we know from geometry that the diameter is twice the radius (a = 2r), the radius of the circumscribing circle is r. The center of this circle is at the midpoint of the base a in the given symmetric form. Therefore, the center of the circle is at the origin due to the symmetric property and the radius remains r.

Answer:

a) There is a probability of 42% that the person will feel guilty for only one of those things.

b)There is a probability of 46% that a randomly selected person will not feel guilty for either of these reasons

Step-by-step explanation:

This probability problem can be solved by building a Venn like diagram for each probability.

I say that we have two sets:

-Set A, for those people that will feel guilty about wasting food.

-Set B, for those people that will feel guilty about leaving lights on when not in a room.

The most important information is that there is a .12 probability that a randomly selected person will feel guilty for both of these reasons. It means that [tex]P(A \cap B) = .12.[/tex]

The problem also states that there is a .39 probability that a randomly selected person will feel guilty about wasting food. It means that P(A) = 0.39. The probability of a person feeling guilty for only wasting food is PO(A) = .39-.12 = .27.

Also, there is a .27 probability that a randomly selected person will feel guilty about leaving lights on when not in a room. So, the probability of a person feeling guilty for only leaving the lights on is PO(B) = 0.27-0.12 = 0.15.

a) What is the probability that a randomly selected person will feel guilty for either wasting food or leaving lights on when not in a room?

This is the probability that the person feels guilt for only one of those things, so:

P = PO(A) + PO(B) = 0.27 + 0.15 = 0.42 = 42%

b) What is the probability that a randomly selected person will not feel guilty for either of these reasons

The sum of all the probabilities is always 1. In this problem, we have the following probabilies

- The person will not feel guilty for either of these reasons: P

- The person will feel guilty for only one of those things:  PO(A) + PO(B) = 0.42

- The person will feel guilty for both reasons: PB = 0.12

So

`P + 0.42 + 0.12 = 1

P = 1-0.54

P = 0.46

There is a probability of 46% that a randomly selected person will not feel guilty for either of these reasons

Answer:

The nth term in the sequence is given by the equation:

[tex]n_{th} =(n+1)^{n-1}[/tex]

Step-by-step explanation:

Arranging a table for n and nth:

[tex]\left[\begin{array}{ccc}n&nth\\1&1\\2&3\\2&16\\4&125\\5&1296\end{array}\right][/tex]

It is easier to notice that 16 and 125 result from the second power of 4 and the third of 5, respectively, which are one number below their respective position. That is why we can deduce that the base of the power is n+1.

For n=2, the base n+1 results in 3, which matches the nth term for n=2. Since 3 is the result of 3 to the power of 1, 16 is 4 to the power of 2, and 125 is 5 to the power of 3, all the powers are one number behind n, so the power is given by n-1, giving the equation:

[tex]n_{th} =(n+1)^{n-1}[/tex]

Answer:

The answer is : 0.0597

Step-by-step explanation:

Population mean = μ = 80

Standard deviation = σ = 20

Sample = N = 60

σ_mean = σ/√N

= (20)/√(60) = 2.582

Now we will find z1 and z2.

z1 = {(84) - μ}/σ_mean = [tex]{(84)-(80)}/(2.582)= 1.549[/tex]

z2 = {(88) - μ}/σ_mean = [tex]{(88)-(80)}/(2.582)= 3.098[/tex]

Now probability that the sample mean will be between 84 and 88 is given by:

Prob{ (1.549) ≤ Z≤ (3.098) } = (0.9990) -(0.9393) = 0.0597