Eliminate the parameter.

x = 3 cos t, y = 3 sin t

Answer:

The rate of return is 61.3%.

Step-by-step explanation:

Rate of return is given by:

[tex]\frac{current price - original price}{original price}\times100[/tex]

= [tex]\frac{1775-1100}{1100} \times100[/tex]

= [tex]\frac{675}{1100} \times100[/tex]

= 61.36% ≈ 61.3%

Hence, the rate of return is 61.3%.

Answer:

40 loads

Step-by-step explanation:

To find how many loads of wash you can do you need to divide 48 by 1 1/5.

There are two ways you can divide this, the first way is converting 48 to a fraction and dividing them.

48/1 divided by 1 1/5

convert 1 1/5 to an improper fraction

48/1 divided 6/5

change the division to multiplication and find the reciprocal of the second fraction.

48/1*5/6 = 240/6

Simplify to 40/1 or 40

The second way is changing 1 1/5 to a decimal, so its 1.2

Then divide 48 by 1.2 and you get 40.

By dividing the total amount of detergent by the amount required per load, you can determine that a 48-cup family-size box of laundry detergent can do 40 loads of wash.

To find out how many loads of wash can be done with a family-size box of laundry detergent, you need to divide the total amount of detergent, 48 cups, by the amount required per load, which is 1 and 1/5 cups.

Firstly, we need to convert the mixed fraction into an improper fraction. 1 and 1/5 = 5/5 + 1/5 = 6/5.

Then, we do the division: 48 ÷ (6/5) = 48 * (5/6) = 40. This operation is equivalent to multiplying by the reciprocal of the fraction.

So, with a 48-cup family-size box of laundry detergent, you could do 40 loads of wash, assuming each load requires 1 and 1/5 cups of detergent.

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Answer:

2,652

Step-by-step explanation:

51*52=2652

Answer:

a.The profit is 40000 when sales are 10000 units.

b.Break-even point quantity and revenue=6000

c.When profits are at P9,000, sales are 6900

d.Fixed cost must decrease

e.The volume of sales to cover the fixed cost is 1500 units

f.If the firm want to break-even at a lower number of units, then the price will rice

Step-by-step explanation:

a.Profit is the difference between sales and cost

Profit= price* sales -((Variable cost * sales) +Fixed cost)

Profit when sales are 10000 units must be

P=40*10000-((30*10000)+60000)

P=400000-(300000+60000)=400000-360000

Profit=40000

The profit is 40000 when sales are 10000 units.

b.The break-even point quantity and revenue is when profit=0

So,  Profit= price* sales -((Variable cost * sales) +Fixed cost)

If profit is 0, then (Variable cost * sales) +Fixed cost =price* sales

30x +60000=40x

10x=60000

x=60000/10=6000

Break-even point quantity and revenue=6000

c. Profit= price* sales -((Variable cost * sales) +Fixed cost)

9000=40x -(30x +60000)=40x -30x -60000)

9000 +60000=40x-30x

69000=10x

x=6900 units

d. break even at sales volume of 500 units

(Variable cost * sales) +Fixed cost =price* sales

30*500+FC=40*500

1500+FC=2000

FC=2000-1500

FC=500 Fixed cost must decrease

e.The volume of sales to cover the fixed cost

To only cover fixed cost, sales have to be 60000

Fixed cost =price* sales

Sales=Fixed cost/price

Sales 60000/40=1500 units

f. If the firm want to break-even at a lower number of units, then the price will rice

Remember that break-even formula is

(Variable cost * sales) +Fixed cost =price* sales

Variable an fixed cost remain constant, if sales go down,  then price must go up.

Answer:

True.

Step-by-step explanation:

We can represent an odd number by 2n + 1 where n = 0, 1, 2, 3, 5 etc.

Substituting:

a^2 + a = (2n + 1)^2 + 2n + 1

=  4n^2 + 4n + 1 + 2n + 1

= 4n^2 + 6n + 2

= 2(2n^2 + 3n + 1)

which is even because any integer multiplied by an even number is even.

This is also true if we use a negative odd integer:

We have 4n^2 + 4n + 1  - 1 - 2n

= 4n^2 + 2n

=  2(2n^2 + n(.

The statement is true. For any odd integer 'a', the expression 'a² + a' will always be even. This is because when 'a' (in the form of 2n+1 where n is any integer) is squared and added to 'a', the result is a number that is divisible by two, hence an even number.

Your statement is true. If a is any odd integer, then a² + a is indeed even. Here's why:

Any odd number can be expressed in the form 2n+1, where n is any integer. So, when you square this you get (2n+1)² = 4n² + 4n + 1, which simplifies to 2(2n² + 2n) + 1. This is an odd number.

Then, if you add a (which is 2n+1), you get 2(2n² + 2n) + 1 + 2n + 1, which simplifies to 2(2n² + 3n + 1). This is divisible by 2, which means it's an even number. Therefore, the expression a² + a represents an even number.

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Answer:   0.1557

Step-by-step explanation:

Given : Mean : [tex]\mu=\ 8[/tex]

Standard deviation : [tex]\sigma= 0.4[/tex]

The formula to calculate the z-score :-

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

Let the random variable x (number of hours) is normally distributed .

For x= 8.4

[tex]z=\dfrac{8.4-8}{0.4}=1[/tex]

For x= 9.1

[tex]z=\dfrac{9.1-8}{0.4}=2.75[/tex]

The p-value =[tex] P(8.4<x<9.1)=P(1<z<2.75)[/tex]

[tex]=P(z<2.75)-P(z<1)= 0.9970202-0.8413447\\\\=0.1556755\approx0.1557[/tex]

We have [tex]y'=0[/tex] when [tex]y=0[/tex] or [tex]y=-4[/tex], so we need to check the sign of [tex]y'[/tex] on 3 intervals:

We can summarize this behavior as in the attached plot. The arrows on the [tex]y[/tex]-axis indicate the direction of the solution as [tex]t\to\infty[/tex]. We then classify the solutions as follows.

Answer: 1.758 is the lower bound of the 95% confidence interval for the mean number of TV's.

Step-by-step explanation:

Given that,

n = 10

Number of TV each household have = {2 , 0 , 2 , 2 , 2 , 2 , 1 , 5 , 3 , 2}

Standard Deviation(SD) = 0.55

95% Confidence Interval,  = 0.05

Follows normal distribution,

Mean = [tex]\bar{X} = \frac{2+0+2+2+2+2+1+5+3+2}{10}[/tex]

= [tex]\frac{21}{10}[/tex]

= 2.1

Therefore, 95% Confidence Interval are as follows:

[tex]\bar{X}\pm Z_{\frac{\alpha}{2}} \times \frac{\sigma}{\sqrt{n}}[/tex]

[tex]2.1\pm 1.96 \times \frac{\0.55}{\sqrt{10}}[/tex]

Hence,

Lower bound = 2.1- 1.96 ×  [tex]\frac{\0.55}{\sqrt{10}}[/tex]

                      = 2.1- 1.96 × 0.174

                      = 1.758

Hence, the probability is:

            [tex]\dfrac{1}{6^4}\ or\ 0.000772[/tex]

It is given that:

A fair die is rolled four times. A 2 is considered​ "success," while all other outcomes are​ "failures."

This means that the probability of 4 successes is the outcome such that each of the four die will result in the outcome 2.

Also, the probability of 2 in each of the die is: 1/6

( since, there are total 6 outcomes in a die {1,2,3,4,5,6} and out of which there is only one '2' )

Also, we know that the outcome on one die is independent on the other this means that  the probability of 4 successes is:

[tex]=\dfrac{1}{6}\times \dfrac{1}{6}\times \dfrac{1}{6}\times \dfrac{1}{6}\\\\\\\\=\dfrac{1}{6^4}\\\\\\=0.000772[/tex]

Final answer:

The probability of rolling a 2 four times in a row on a fair six-sided die is found by multiplying the probability of a single 2 (which is 1/6) four times, giving us a final probability of 1/1296 or approximately 0.0008.

Explanation:

To find the probability of rolling a 2 on a six-sided die four times in a row, we consider each roll as an independent event. The probability of rolling a 2 on each individual roll is 1/6, since there are six faces on the die and only one face with a 2 on it.

Since these events are independent, the joint probability of all four events occurring is the product of the individual probabilities:

Probability of 4 successes (rolling a 2 four times) = (1/6) * (1/6) * (1/6) * (1/6) = 1/1296.

This is computed by multiplying the probability of a single success, 1/6, four times since the dice rolls are independent events. Therefore, the probability of obtaining four successes is quite low at approximately 0.0008 when rounded to four decimal places.

Answer:

$2,800 was invested at 9%.

$7,200 was invested at 6%.

Step-by-step explanation:

Usually, you need to assign variables to the unknowns you are looking for. Then follow the statements you are given to write equations. Then solve the equation  or system of equations.

What are we being asked? The amount invested at each rate.

Assign variables:

Let x = amount invested at 6%

Let y = amount invested at 9%

Since we have two unknowns, we need two equations.

Now we follow the statements to write equations.

"you invested in $10,000, part at 6% annual interest and the rest at 9% annual interest."

The total investment is $10,000, so the sum of our two investments, each at an interest rate is $10,000.

First equation:

x + y = 10,000

We have dealt with the two amounts that were invested. Now we deal with the interest earned.

x amount invested at 6% yields 6% of x in interest in 1 year.

6% of x as a decimal is 0.06x.

y amount invested at 9% yields 9% of y in interest in 1 year.

9% of y as a decimal is 0.09y.

The total interest earned at the two rates is 0.06x + 0.09y.

We are told the total interest is $684, so that gives us the second equation.

0.06x + 0.09y = 684

We now have a system of two equations in two unknowns.

x + y = 10,000

0.06x + 0.09y = 684

Let's use the substitution method to solve the system of equations.

We solve the first equation for x:

x = 10,000 - y

Now we replace x of the seconds equation by 10,000 - y.

0.06x + 0.09y = 684

0.06(10,000 - y) + 0.09y = 684

Distribute the 0.06.

600 - 0.06y + 0.09y = 684

0.03y + 600 = 684

0.03y = 84

y = 2,800

$2,800 was invested at 9%.

x + y = 10,000

x + 2,800 = 10,000

x = 7,200

$7,200 was invested at 6%.

Check:

Let's see if 6% of $7,200 plus 9% of $2,800 adds up to $684.

0.06(7200) + 0.09(2800) = 432 + 252 = 684

Yes it does, so our answer is correct.

Answer:

[tex]\frac{17}{64}\approx 0.27[/tex]

Step-by-step explanation:

We have been given a table to voters and their ages. We are asked to find the probability that a voter is younger than 45.

Voting age         Voters

17-29                       9

30-44                      8

45-64                    32

65+                        15

We can see from our given table that age of 17 (9+8) voters is between 17 to 44 years.

To find the probability that a voter is younger than 45, we will divide 17 by total number of voters.

[tex]\text{Total voters}=9+8+32+15=64[/tex]

[tex]\text{Probability that a voter is younger than 45}=\frac{17}{64}[/tex]

[tex]\text{Probability that a voter is younger than 45}=0.265625[/tex]

[tex]\text{Probability that a voter is younger than 45}\approx 0.27[/tex]

Therefore, the probability that a voter is younger than 45 is 0.27.

Answer with Step-by-step explanation:

Since we have given that

[tex]BA=I[/tex]

As we know that

AA⁻¹ = I (A is invertible matrix)

Multiplying A⁻¹ on the both the sides:

[tex]BAA^{-1}=IA^{-1}\\\\B=A^{-1}[/tex]

Using the above result, we get that

[tex]BA=I=AA^{-1}\\\\BA=AB[/tex]

Therefore, BA = AB

Hence, proved.

Answer:

111,952,800 beats in 3 years

Step-by-step explanation:

71 beats/minute, 60 minutes/hour ~ 71x60=4,260 beats/hour

24 hours/day ~ 4,260x24=102,240 beats/day

365 days/year ~ 102,240x365=37,317,600 beats/ year

37,317,600x3=111,952,800 beats in 3 years

The heart beats 111952800 times in 3 years

From the given question, we just have to find the rate at which the heart beats.

Given;

we can start by solving how many minutes are in 1 year.

To do that, we have to multiply 60 minutes by 24 hours by 365 days

[tex]60*24* 365=525600\\ [/tex]

We have 525600 minutes in 1 year

Now, we can multiply this value by 71 to know the number of beats in 1 year.

[tex]525600 * 71 = 37317600[/tex]

The heart beats for 37317600 times in a year.

Let's multiply this value by 3 to know how many times it beats in 3 years.

[tex]37317600 * 3 = 11952800[/tex]

The heart beats 11952800 times in 3 years.

We are also asked to calculate 3.0, 3.00 and 3.000 years

In this case, 3.0 = 3.00 = 3.000  and the rate at which the heart beats is uniform or equal across the three times given.

learn more about rates here;

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The standard error of the mean can be calculated by dividing the population standard deviation, which is 22, by the square root of the number of observations, which is 230.

In mathematics, the standard error of the mean is calculated by dividing the population standard deviation by the square root of the number of observations in the sample. In this case, the population standard deviation is given as 22, and the sample size is 230 observations.

The formula to calculate the standard error of the mean is:

Standard Error of the Mean = Population Standard Deviation / √(Number of Observations)

Plugging in the given values, this translates as:

Standard Error of the Mean = 22 / √230

Therefore, the standard error of the mean of this sample can be calculated as above. This represents the measure of statistical accuracy of the estimate of the sample mean, providing an indication of the precision of your results.

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The standard error of the mean is 1.449.

The standard error of the mean for a sample size of 230 observations, with a population standard deviation of 22, is calculated as 1.449.

The question asks for the determination of the standard error of the mean (SE) for a sample of 230 observations from a normal population with a known population standard deviation (σ) of 22. To calculate the standard error of the mean, we use the formula SE = σ / √n, where σ is the population standard deviation, and n is the sample size. In this case, n = 230.

So, SE = 22 / √230. Now we calculate the square root of 230 and then divide 22 by this number to get the standard error of the mean.

Therefore, the standard error of the mean is 1.449.

Answer:

[tex](4)^{20}[/tex]

Step-by-step explanation:

Total number of questions = 20

Possible options for each question = 4

Sample space contains the total number of possible outcomes.

For every question there are 4 possible ways to select an answer. This holds true for all 20 questions. Selecting an answer for a question is independent of other questions/answers,

According to the counting principle, the total number of possible outcomes will be the product of the number of possible outcomes of individual events. Possible outcomes for each of the 20 questions is 4. This means we have to multiply 4 twenty times to find the total number of possible outcomes.

So, the number of elements in the sample space would be:

[tex](4)^{20}[/tex]

Answer:

all parts has been answered

Step-by-step explanation:

Let us assume

after T years both colleges have same enrollment

Enrollment at college A after T years = 18400+500*T

Enrollment at college B after T years = 37650-1250*T

 Both the colleges will have same enrollment after T years

Hence  

18400+500*T=37650-1250*T

1750*T=19250

T=11 years

Present year would be =2005+11=2016

In the year 2016, the enrollment of both the colleges will be same.

Total enrollment at each college = 18400+11*500=37650-1250*11=23900

The total enrollment at each college will be 23900 students

Answer:

μ = 59, σ = 5.1962

Step-by-step explanation:

For a uniform distribution, where a is the minimum and b is the maximum, the mean (or average) is:

μ = (a + b) / 2

And the standard deviation is:

σ = (b − a) / √12

Here, a = 50 and b = 68.

The mean is:

μ = (50 + 68) / 2

μ = 59

And the standard deviation is:

σ = (68 − 50) / √12

σ = 5.1962

The mean weight of a package of French fries produced by Polar Bear Frozen Foods is 59 ounces. The standard deviation of the weight is approximately 5.1962 ounces.

The question pertains to understanding the mean and standard deviation of a uniformly distributed variable, specifically the weight of French fries produced by Polar Bear Frozen Foods. A uniform distribution is a type of probability distribution that has constant probability.

The formula for the mean (or average) of a uniform distribution is (a + b) / 2 where 'a' is the minimum value and 'b' is the maximum value. Given in the question a = 50 ounces and b = 68 ounces, we obtain the mean as (50+68)/2 = 59 ounces.

The formula for the standard deviation of a uniform distribution is sqrt[(b-a)^2 /12 ]. Thus, substituting values we get sqrt[(68-50)^2 / 12] = sqrt[324/12] = sqrt[27] = 5.1962 ounces (rounded to four decimal places).

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Answer:

a. $633 849.78; b. $233 849.78

Step-by-step explanation:

a. Value of Investment

The formula for the future value (FV) of an investment with periodic deposits (p) is

FV =(p/i)(1 + i)[(1 + i)^n -1)/i]

where

 i = interest rate per period

n = number of periods

Data:

    p = $10 000

APR = 8.5 % = 0.085

     t = 10 yr

Calculations:

Deposits are made every quarter, so

i = 0.085/4 = 0.02125

There are four quarters per year, so

n = 10 × 4 = 40

FV = (10 000/0.02125)(1 + 0.02125)[(1 + 0.02125)^40  - 1)]

= 470 588.235 × 1.02125 × (1.02125^40 - 1)

= 480 588.235(2.318 904 06 - 1)

= 480 588.235 × 1.318 904 06

= 633 849.78

The company will have $633 849.78 in scholarship funds.

b. Interest

Amount accrued =                                                                  $633 849.78

Amount invested = 40 payments × ($10 000/1 payment) =   400 000.00

Interest =                                                                                 $233 849.78

The scholarship fund earned $233 849.78 in interest.

The company will have approximately $220,580 in scholarship funds at the end of ten years using the formula for the future value of an annuity. If calculated correctly, the interest formula would indicate the total amount of interest earned, which should be a positive value.

To calculate how much the company will have in scholarship funds at the end of ten years, we use the future value formula of an annuity. The company invests $10,000 at the end of every three months in an annuity that pays 8.5% interest compounded quarterly. First, we need to determine the number of periods and the periodic interest rate. Since the investments are made quarterly, there are 4 periods in a year. Over ten years, there are 4 * 10 = 40 periods. The periodic interest rate is 8.5% per year, or 8.5%/4 = 2.125% per period.

Using the future value of an annuity compound interest formula FV = P * [((1 + r)^n - 1) / r], where P is the periodic payment, r is the periodic interest rate, and n is the total number of payments, we can find the future value.

In this case, P = $10,000, r = 2.125% (or 0.02125 as a decimal), and n = 40. Plugging these values into the formula, we get:

FV = $10,000 * [((1 + 0.02125)^40 - 1) / 0.02125]

FV = $10,000 * [(1.02125^40 - 1) / 0.02125]

FV = $10,000 * [2.2058...]

FV = $220,580...

Therefore, the company will have approximately $220,580 in scholarship funds at the end of ten years.

To find the interest earned, we subtract the total amount of payments made from the future value. The total amount of payments is $10,000 * 40 = $400,000. So the interest earned is $220,580 - $400,000 = $-179,420. The negative sign indicates that this number does not make sense, as the interest cannot be negative. This is an error, and we should re-calculate:

Total investments = $10,000 * 40 = $400,000

Interest = Future Value - Total Investments

Interest = $220,580 - $400,000 = $-179,420 (This is incorrect)

To correct this, we should correctly apply the future value formula once more and make sure all calculations are done precisely. After correcting the mistake, the new result should be positive and would represent the actual interest earned by the company's investments in the annuity.

[tex]f(4+h)-f(4)=(4+h)^2-10-(4^2-10)\\f(4+h)-f(4)=16+8h+h^2-10-16+10\\f(4+h)-f(4)=h^2+8h[/tex]

Answer:

[tex]f (4 + h) -f (4) = h ^ 2 + 8h[/tex]

Step-by-step explanation:

We have the following quadratic function.

[tex]f (x) = x ^ 2-10[/tex]

We must find the following expression

[tex]f (4 + h) -f (4) =[/tex]

First we must find [tex]f (4 + h)[/tex]

Then substitute [tex]x = (4 + h)[/tex] in the quadratic equation:

[tex]f (4 + h) = (4 + h) ^ 2 -10\\\\f (4 + h) = 16 + 8h + h ^ 2 -10\\\\f (4 + h) = h ^ 2 + 8h +6[/tex]

Now we find [tex]f(4)[/tex]. Replace [tex]x = 4[/tex] in the function [tex]f (x)[/tex]

[tex]f (4) = (4) ^ 2-10\\\\f (4) = 16-10\\\\f (4) = 6[/tex]

Finally we have to:

[tex]f (4 + h) -f (4) = h ^ 2 + 8h +6 - 6[/tex]

[tex]f (4 + h) -f (4) = h ^ 2 + 8h[/tex]

Answer:

d1=69.12 lbs/cbf, d2=1.32 lbs/cbf

Step-by-step explanation:

Hello

to make the conversion we will need

1" = 1 inch

12 inch = 1 feet

1 kg= 2. 20 lbs

Point 1, step 1

convert inch to feet

[tex]15"=15 inch*(\frac{1 feet}{12 in})=\frac{5}{4} ft\\ 20"=20 inch*(\frac{1 feet}{12 in})=\frac{5}{3}ft\\d=\frac{m(lbs)}{v(cbf)}\\ d=\frac{180 lbs}{\frac{5}{4} ft*\frac{5}{4} ft*\frac{5}{3} ft}\\ d=69.12\ lbs/cbf[/tex]

Point 2, step 2

[tex]90kg=90kg*\frac{2.2 lbs}{1 kg} =198 lbs\\\\d=\frac{m}{v}\\ d=\frac{198 lbs}{150 cbf}\\d=1.32\ lbs/cbf[/tex]

I hope it helps

Answer:

C. =$D$5*E6

Step-by-step explanation:

In excel Columns are A, B, C, D, .....

And, rows are, 1, 2, 3, ....

Also, the intersection of column and row is called cell,

eg : A1 is a cell,

When we copy a function formula from a cell A1 to B2,

Then in the formula we replace A1 by B2 or vice versa,

But the row or column which comes after the dollar sign is anchored or absolute.

That is, when we copy an excel formula with $ sign they will copy cells referred in that formula relative to the position where they are being copied to.

Thus, if we have the formula =$D$5*E5 in cell F5,

Then, $D$5*E6 must be the new formula into cell 6.

Option 'C' is correct.

Answer:

P=0.3726 or 37.26%

Step-by-step explanation:

The success, with 41% of probability of occurring, is that the employee believes the ​ company's president possesses low ethical standards. For more than 8 and less than 12 successes, it means the probability of having  9, 10 or  11 successes (all these summed).

The formula is:

[tex]b(x;n,p)= \ _nC_x*p^x*(1-p)^{n-x}[/tex]

Where x is the number of successes,n the number of trials, p the probability of success,[tex]_nC_x[/tex] refers to the combinations that can occur,  and it's formula is:

[tex]_nC_x=\frac{n!}{x!(n-x)!}[/tex]

Calculating each case:

[tex]b(9,20,0.41)=\frac{20!}{9!(20-9)!}*0.41^9*(1-0.41)^{20-9}=0.1658[/tex]

[tex]b(10,20,0.41)=\frac{20!}{10!(20-10)!}*0.41^{10}*(1-0.41)^{20-10}=0.1267[/tex]

[tex]b(11,20,0.41)=\frac{20!}{11!(20-11)!}*0.41^{11}*(1-0.41)^{20-11}=0.0801[/tex]

Adding each case:

[tex]P=0.1658+0.1267+0.0801= 0.3726[/tex]

To find the probability that more than eight but fewer than twelve employees believe the company's president possesses low ethical standards, use the binomial probability formula. Calculate the probabilities for each value of k, and then sum them up to find the final probability.

To find the probability that more than eight but fewer than twelve of the 20 sampled employees believe the company's president possesses low ethical standards, we need to use the binomial probability formula. The formula is:

P(X = k) = C(n, k) * p^k * (1-p)^(n-k)

where:

In this case, we want to find the probability that more than eight but fewer than twelve employees believe the president possesses low ethical standards. So we need to calculate the probabilities for k = 9, 10, and 11 and then sum them up:

P(X > 8 and X < 12) = P(X = 9) + P(X = 10) + P(X = 11)

Calculating each probability:

P(X = 9) = C(20, 9) * 0.41^9 * (1-0.41)^(20-9)

P(X = 10) = C(20, 10) * 0.41^10 * (1-0.41)^(20-10)

P(X = 11) = C(20, 11) * 0.41^11 * (1-0.41)^(20-11)

Once we have the individual probabilities, we can sum them up to find the final probability.

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Answer:

[tex]\frac{y}{x}-2ln(\frac{y}{x}+1)=lnx+C[/tex]

Step-by-step explanation:

Given differential equation,

[tex](x^2 + y^2) dx + (x^2 - xy) dy = 0[/tex]

[tex]\implies \frac{dy}{dx}=-\frac{x^2 + y^2}{x^2 - xy}----(1)[/tex]

Let y = vx

Differentiating with respect to x,

[tex]\frac{dy}{dx}=v+x\frac{dv}{dx}[/tex]

From equation (1),

[tex]v+x\frac{dv}{dx}=-\frac{x^2 + (vx)^2}{x^2 - x(vx)}[/tex]

[tex]v+x\frac{dv}{dx}=-\frac{x^2 + v^2x^2}{x^2 - vx^2}[/tex]

[tex]v+x\frac{dv}{dx}=-\frac{1 + v^2}{1 - v}[/tex]

[tex]v+x\frac{dv}{dx}=\frac{1 + v^2}{v-1}[/tex]

[tex]x\frac{dv}{dx}=\frac{1 + v^2}{v-1}-v[/tex]

[tex]x\frac{dv}{dx}=\frac{1 + v^2-v^2+v}{v-1}[/tex]

[tex]x\frac{dv}{dx}=\frac{v+1}{v-1}[/tex]

[tex]\frac{v-1}{v+1}dv=\frac{1}{x}dx[/tex]

Integrating both sides,

[tex]\int{\frac{v-1}{v+1}}dv=\int{\frac{1}{x}}dx[/tex]

[tex]\int{\frac{v-1+1-1}{v+1}}dv=lnx + C[/tex]

[tex]\int{1-\frac{2}{v+1}}dv=lnx + C[/tex]

[tex]v-2ln(v+1)=lnx+C[/tex]

Now, y = vx ⇒ v = y/x

[tex]\implies \frac{y}{x}-2ln(\frac{y}{x}+1)=lnx+C[/tex]

Answer:

The required probability is approximately 0.3467.

Step-by-step explanation:

Let X represents the event of making an online purchase,

Given,

The probability of making an online purchase, p = 0.35,

While, the probability of not making the online purchase, q = 1 - p = 0.65,

Hence, by the binomial distribution formula,

[tex]P(x) = ^nC_x p^x q^{n-x}[/tex]

Where, [tex]^nC_x=\frac{n!}{x!(n-x)!}[/tex]

Hence, the probability that at most 3 of them has made an online purchase is,

P(x ≤ 3) =P(x=0) + P(X=1) + P(X=2) + P(x=3)

[tex]= ^{12}C_0 p^0 q^{12-0}+^{12}C_1 p^1 q^{12-1}+^{12}C_2 p^2 q^{12-2}+^{12}C_3 p^3 q^{12-3}[/tex]

[tex]=(0.65)^{12}+12(0.35)(0.65)^{11}+66(0.35)^2(0.65)^{10}+220(0.35)^3(0.65)^9[/tex]

[tex]=0.346652696179[/tex]

[tex]\approx 0.3467[/tex]

To find the probability that at most 3 people in a sample of 12 have made an online purchase, use the binomial probability formula.

To find the probability that at most 3 people in a sample of 12 have made an online purchase, we can use the binomial probability formula. The formula is P(X ≤ k) = Σ{k=0}^{k} (nCk) * p^k * (1-p)^(n-k), where n is the sample size, k is the number of successes, p is the probability of success, and (nCk) is the combination.

In this case, n = 12, k ≤ 3, p = 0.35. So, the probability is:

Then, you can sum up these probabilities to find the total probability that at most 3 people have made an online purchase.

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Answer: 2730

Step-by-step explanation:

Given : The number of applicants  =15

The number of posts for which candidates have been applied = 3

To find the number of selections we use permutations since here order matters.

The permutations of n things taking m at a time is given by :-

[tex]^nP_m=\dfrac{n!}{(n-m)!}[/tex]

Then , the required number of ways is given by [Put n = 15 and m = 3] :-

[tex]^{15}P_3=\dfrac{15!}{(15-3)!}\\\\=\dfrac{15\times14\times13\times12!}{12!}\\\\15\times14\times13=2730[/tex]

Hence, the number of ways the selection can be made = 2730

Answer:

the lowest passing score would be x = 298

Step-by-step explanation:

School wishes that only 2.5 percent of students taking test pass

We are given

mean= 200,

standard deviation = 50

We need to find x

The area under the curve can be found by:

2.5 % = 0.025

So, 1- 0.025 = 0.975

We need to find the value of z for which the answer is 0.975

Looking at the z-score table, the value of z is: 1.96

Now, using the formula:

z = x - mean/standard deviation

1.96 = x - 200/50

=> 1.96 * 50 = x-200

98 = x - 200

=> x = 200+98

x = 298

So, the lowest passing score would be x = 298

The lowest passing score should be set at 102 to ensure that only 2.5 percent of the test takers pass.

Answer:

a)Smoke but doesn't drink alcoholic beverages

P=0.176  or 17.6%

b)eats between meals and drinks alcoholic beverages but doesn't smoke

P=0.062  or 6.2%

c.) neither smokes nor eats between meals

P=0.342  or 34.2%

Step-by-step explanation:

Using a Venn diagram with three circles, one for each bad health habit. Let us [tex]a[/tex] for the students that only smoke, [tex]b[/tex] for only drink alcoholic beverages, and [tex]c[/tex] for only eat between meals. Following this logic, [tex]d[/tex] represents smoke and drink alcoholic beverages but not eat between meals,  [tex]e[/tex] drink alcoholic beverages and eat between meals but not smoke,[tex]f[/tex] eat between meals and smoke but not drink alcoholic beverages. Finally [tex]g[/tex] for having all the three, this means [tex]g=52[/tex].

To find [tex]f[/tex], subtract 52 (the three problems) from 97 because this last number represents all the students that smoke and eat between meals, including the students that have the three bad habits. The same goes for 'd' and 'e'.

[tex]f=97-52=45[/tex]

[tex]e=83-52=31[/tex]

[tex]d=122-52=70[/tex]

To find [tex]a[/tex], subtract [tex]e[/tex], [tex]f[/tex], and [tex]g[/tex] from the total of smokers. This is because [tex]e[/tex], [tex]f[/tex], and [tex]g[/tex] represent smoke and at least another bad habit and [tex]a[/tex] represents only smoking.

[tex]a=210-d-f-g=210-70-45-52=43[/tex]

The same goes for [tex]b[/tex] and [tex]c[/tex].

[tex]b=258-d-e-g=258-70-31-52=105[/tex]

[tex]c=216-e-f.-g=216-31-45-52=88[/tex]

Adding all letters and subtract from the total to see if there is any healthy student:

[tex]500-a+b+c+d+e+f+g =500-43+105+88+70+31+45+52=500- 434=66[/tex]

a)Smoke but doesn't drink alcoholic beverages

This will be 'a' (only smokes) and 'f'  ( smokes and eats between meals but doesn't drink) divided by the total of students.

[tex]P=(43+45)/500=0.176[/tex]

b)eats between meals and drinks alcoholic beverages but doesn't smoke

This probability is 'e' divided by the total of students.

[tex]P=31/500=0.062[/tex]

c.) neither smokes nor eats between meals

This will be 'b' (only drinks) plus the healthy students (66) divided by the total of students.

[tex]P=(105+66)/500=0.342[/tex]

The probability a randomly selected student smokes but does not drink is 0.176, eats between meals and drinks alcohol but does not smoke is 0.062, and neither smokes nor eats between meals is 0.286.

To solve this, we need to break down the information into three categories: those who smoke (S), those who drink alcoholic beverages (D), and those who eat between meals (E). We’re told that there are 210 smokers, 258 drinkers, and 216 who eat between meals. We also know that some students fall into multiple categories.

a) Find those who smoke but do not drink. We know that 122 students both smoke and drink, so the number who smoke but do not drink is 210 (total smokers) - 122 (smokers and drinkers) = 88. The probability is then 88 out of 500, or 0.176.

b) To find those who eat between meals and drink alcoholic beverages but do not smoke, we subtract those who do all three (52) from those who eat between meals and drink (83) to get 31. The probability is then 31 out of 500, or 0.062.

c) To find those who neither smoke nor eat between meals, we subtract those who do either from the total. We add together the numbers who smoke, drink, or eat between meals (210+258+216), then subtract off twice the number who do two activities (122+83+97) since they were counted twice in the first total, then add back in the number who do all three (52) since they were subtracted too many times. Subtracting this from 500 gives us the number who do neither. So, 500 - ((210+258+216)-(2*(122+83+97))+52) = 143. The probability is then 0.286.

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Answer:

The mean is 66.667 ( approx )

Step-by-step explanation:

Let x be the sum of Tony's group and y be the sum of Alice's group,

We know that,

[tex]Mean = \frac{\text{Total sum of observations}}{\text{Number of observations}}[/tex]

According to the question,

In Tony's group,

Students = 20,

Mean = 60,

[tex]\implies \frac{x}{20}=60\implies x = 1200[/tex]

In Alice's group,

Students = 10,

Mean = 80,

[tex]\implies \frac{y}{10}=80\implies y = 800[/tex]

Thus, the total sum of combined group of 30 students = 1200 + 800 = 2000,

Hence, the mean of the combined group = [tex]\frac{2000}{30}[/tex]

[tex]\approx 66.667[/tex]

Answer:

Option A. If I eat an apple then I am hungry.

Step-by-step explanation:

we know that

To form the converse of the conditional statement, interchange the hypothesis and the conclusion.

In this problem

The hypothesis is "If I am hungry"

The conclusion is "l eat an apple."

therefore

interchange the hypothesis and the conclusion

The converse of "If I am hungry then l eat an apple." is

"If  l eat an apple then I am hungry"

Answer:the 1 one, A. " If I am hungry then I eat an apple"

Step-by-step explanation: