The sum of the circumference of a circle and the perimeter of a square is 24. Find the dimensions of the circle and square that produce a minimum total area. (Let x be the length of a side of the square and r be the radius of the circle.)

Answer: The net is made of 6 squares, but one face is open. I would add the area of 5 squares to get the surface area in square feet. Then, I would multiply the area by 4 to find the number of tiles needed to cover the surface area.

Step-by-step explanation:

How to use a net to find the number of tiles needed to cover the entire outer surface of the bird bath is: Multiply the surface area by 4.

Surface area can be defined as the number  of space that  cover the outer surface of a dimensional shape.

Since 4  cover one square foot of area in order to determine the number of tiles to cover the outer surface let the net be 6 squares and the surface area be 5 squares because we have one open face.

Multiply the surface area by the 4 tiles which will give us the number of tiles needed.

Therefore Multiply the surface area by 4.

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Answer:

A

Explanation:

For functions to be inverse, it must be true that f( g(x) ) = x and g( f(x) ) = x.

But for F( G(x) ), we have √( G(x) - 3 ) + 8

= √( (x+8)² - 3 - 3 ) + 8

= √( (x+8)² - 6 ) + 8

This -6 part should be cancelled out for functions to work out but we cannot do that, therefore F(x) and G(x) are not inverse.

The given function is not an inverse function across the domain [3,+∞)

We have given that the functions F(x) and G(x)

We have to determine the functions F(x) and G(x) inverse function across the domain  [3,+∞)

For functions to be inverse, it must be true that f( g(x) ) = x and g( f(x) ) = x.

The inverse function of a function f is a function that undoes the operation of f. The inverse of f exists if and only if f is bijective, and if it exists, is denoted by[tex]{\displaystyle f^{-1}.}[/tex]

But for F( G(x) ), we have √( G(x) - 3 ) + 8

[tex]= \sqrt{( (x+8)^2} ( - 3 - 3 ) + 8= \sqrt ( (x+8)^ {- 6} ) + 8[/tex]

This -6 part should be canceled out for functions to work out but we cannot do that, therefore F(x) and G(x) are not inverse.

Therefore the given function is not an inverse function across the domain [3,+∞).

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Answer: 7.7055

Step-by-step explanation:

Given: Sample size : [tex]n= 38> 30\text{ i.e. Large sample}[/tex]

Sample Mean : [tex]\overline{x}=12.15[/tex]

Standard deviation : [tex]\sigma =0.12[/tex]

Claim : The population mean is [tex]\mu=12[/tex]

We assume the contents of cans of Coke have a normal distribution .

We know that the test-static for population mean for larger sample is given by :-

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

[tex]\Rightarrow z=\dfrac{12.15-12}{\dfrac{0.12}{\sqrt{38}}}=7.70551750371\approx7.7055[/tex]

Hence, the value of the test statistic z for the claim that the population mean is μ=12 is 7.7055.

The calculated z-score is approximately 77.72, for which sample mean = 12.15, standard deviation = 0.12, and population mean =12.

To determine the value of the test statistic z for the given data, follow these steps:

The test statistic z is approximately 77.72, indicating how many standard errors the sample mean is from the population mean.

Answer: 0.1353

Step-by-step explanation:

Given : The mean of failures =  0.025 per hour.

Then  for 8 hours , the mean of failures = [tex]\lambda=8\times0.25=2[/tex] per eight hours.

Let X be the number of failures.

The formula to calculate the Poisson distribution is given by :_

[tex]P(X=x)=\dfrac{e^{-\lambda}\lambda^x}{x!}[/tex]

Now, the probability that the instrument does not fail in an 8-hour shift :-

[tex]P(X=0)=\dfrac{e^{-2}2^0}{0!}=0.1353352\approx0.1353[/tex]

Hence, the the probability that the instrument does not fail in an 8-hour shift = 0.1353

The mid line of a triangle is always half of the triangle's "base"

Plug all the answer choices in and see if they produce a true answer

20 = (2(10) - 8) * 2

20 = (20 - 8) * 2

20 = 12 * 2

20 ≠ 24

10 is NOT x

20 = (2(11) - 8) * 2

20 = (22 - 8) * 2

20 = 14 * 2

20 ≠ 28

11 is NOT x

20 = (2(14) - 8) * 2

20 = (28 - 8) * 2

20 = 20 * 2

20 ≠ 40

14 is NOT x

20 = (2(9) - 8) * 2

20 = (18 - 8) * 2

20 = 10 * 2

20 = 20

9 IS x!!!

Or you can do it this way:

20 = (2x - 8) * 2

Divide 2 to both sides

10 = 2x - 8

18 = 2x

x = 9

Hope this helped!

~Just a girl in love with Shawn Mendes

Answer:

New mean = $40920

New Median = $38500 (Same as old)

Step-by-step explanation:

Given

Mean = $40300

Median = $38500

The formula for mean is:

Mean = Sum/No. of Values

According to the formula,

40300 = Sum/5

40300*5 = Sum

Sum of Salaries = 201500

If the salary of lowest employee is raised 3100, then sum will be 204600.

The new mean will be:

Mean = 204600/5

Mean =$40920

The median will remain unchanged as it is the middle value of 5 quantities. 2 values are greater than median and two are less than median. Even after the raise in salary, the salary becomes $35100 which is still less than median. So the median will be same as old ..

With the raise, the new mean salary of the employees is $40920, while the median salary is likely to remain $38500 unless the raise causes a rearrangement in the salary sequence.

Assuming there have been no other changes to the employees' salaries, we can calculate the new mean and new median as follows:

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Answer: There is probability of 0.367 that there will be no breakdown on his car in the trip.

Step-by-step explanation:

Since we have given that

Mean (λ) = 0.5 breakdown per week

Number of weeks the owner of the car is planning to have a trip on his car for = 2 weeks

So, mean for 2 weeks would be

[tex]0.5\times 2=1.0[/tex]

We need to find the probability that there will be no breakdown on his car in the trip.

Probability that there will be no breakdown on his car in the trip is given by

P(X=0) is given by

[tex]\dfrac{e^{-\lambda}\lambda^k}{k!}\\\\=\dfrac{e^{-1}1^0}{0!}\\\\=0.367[/tex]

Hence, there is probability of 0.367 that there will be no breakdown on his car in the trip.

Answer: [tex]x<2[/tex]

Step-by-step explanation:

Given the inequality [tex]2x + 1 < 5[/tex] you can follow these steps to solve it:

- The first step is:

Subtract 1 from both sides on the inequaltity.

Then:

[tex]2x + 1-(1) < 5-(1)\\\\2x < 4[/tex]

- The second  and final step is:

Divide both sides of the inequality by 2.

Therefore, you get:

[tex]\frac{2x}{2}<\frac{4}{2} \\\\(1)x<2\\\\x<2[/tex]

Answer:

-2.333

Step-by-step explanation:

In a certain normal distribution of scores, the mean is 30 and the standard deviation is 3. Find the z-score corresponding to a score of 23.

The z-score corresponding to an observed value in a normal distribution is calculated as;

z-score = (observed value - mean)/(standard deviation)

Our observed score is 23, the mean is 30, and the standard deviation is 3. The z-score will thus be;

z-score = ( 23 - 30)/( 3)

z-score = -2.333

Answer:

- 2.333

Step-by-step explanation:

The formulae to apply here is;

z= (x-μ) / δ------------where x is the score, μ is the mean and δ is the standard deviation

Given x=23, μ=30 and δ= 3

z= (23-30) / 3 z= - 7/3 z=  -2.333

Answer:it is 18*20 effective every year

Step-by-step explanation:

To find the effective annual rate for an 18% yearly interest compounded daily, the formula (1 + i/n)^(n*t) - 1 is used.

For a yearly rate, 'n' equals 365 and 't' equals 1, while for a semi-annual rate, 't' is 0.5. These calculations show the true interest earnings per period.

To calculate the effective annual rate (EAR) for an 18% per year, compounded daily interest rate, we use the formula for the EAR, which is:

EAR = (1 + i/n)^(n*t) - 1

Where i is the nominal interest rate, n is the number of compounding periods per year, and t is the number of years. For daily compounding, n = 365, as there are 365 days in a year.

The calculation for a yearly (annual) effective rate would be:

EAR = (1 + 0.18/365)^(365*1) - 1

For a semi-annual effective rate, we consider the compounding effect over half a year, so t = 0.5:

EAR semi-annual = (1 + 0.18/365)^(365*0.5) - 1

These calculations give you the effective rate of interest for each compounding period.

[tex]\bf \sqrt[4]{16}\implies \sqrt[4]{2^4}\implies 2~\hspace{10em}\sqrt[8]{256}\implies \sqrt[8]{2^8}\implies 2[/tex]

Using the binomial distribution, the professor should expect 101 students to admit to binge drinking if 40% is accurate. With 92 admitting it, and without performing a hypothesis test, the result is not necessarily surprising and could be within statistical fluctuations.

The question at hand involves determining whether a professor should be surprised that 92 out of 252 students surveyed admitted to binge drinking. To answer this, we can use the binomial probability distribution to see if the observed proportion significantly differs from the claimed proportion.

A claimed proportion of 40% would expect 40.2% of 252 students, or about 101 students, to admit to binge drinking. The professor found that only 92 students out of 252 admitted to binge drinking, which is less than expected.

To assess whether this result is surprising, we would perform a hypothesis test for the proportion with the null hypothesis being that the true proportion of binge drinkers is 40%. The alternative hypothesis could be that the true proportion is not 40%, which would require a two-tailed test. Calculations for the z-score and the corresponding p-value would provide the necessary evidence to determine if the professor should be surprised. However, with such results, a minor deviation like this may not be considered statistically significant without further testing.

Answer:  3

Step-by-step explanation:

Let E be the event of that student pierces ear and N be the event of that student pierces nose.

Given: [tex]n(E\cup N=50)[/tex]

[tex]n(E)=46\\\\n(N)=7[/tex]

For any two event A and B, we have

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

Similarly , [tex]n(E\cup N)=n(E)+n(N)-n(E\cap N)[/tex]

[tex]50=46+7-n(E\cap N)\\\\\Rightarrow\ n(E\cap N)=53-50=3[/tex]

Hence, 3 students have piercings both on their ears and their noses.

The volume of [tex]E[/tex] is

[tex]\displaystyle V(E)=\iiint_E\mathrm dV[/tex]

To compute the integral, convert to cylindrical coordinates:

[tex]x=r\cos\theta[/tex]

[tex]y=r\sin\theta[/tex]

[tex]z=z[/tex]

[tex]\implies\mathrm dV=r\,\mathrm dr\,\mathrm d\theta\,\mathrm dz[/tex]

[tex]\displaystyle V(E)=\int_0^{2\pi}\int_0^3\int_0^{9-r^2}r\,\mathrm dz\,\mathrm dr\,\mathrm d\theta=\frac{81\pi}2[/tex]

Now integrate [tex]f[/tex] over [tex]E[/tex]. In cylindrical coordinates, we get

[tex]\displaystyle\iiint_E3x^2z+3y^2z\,\mathrm dV=3\int_0^{2\pi}\int_0^3\int_0^{9-r^2}r^3z\,\mathrm dz\,\mathrm dr\,\mathrm d\theta=\frac{6561\pi}8[/tex]

Then the average value of [tex]f[/tex] over [tex]E[/tex] is [tex]\dfrac{\frac{6561\pi}8}{\frac{81\pi}2}=\dfrac{81}4[/tex].

The average value of a function over a certain region can be found by integrating the function over the volume and then dividing by the volume. For the given function and region, one would integrate over the range of values that satisfy the inequality z = 9 - [tex]x^2 - y^2[/tex] >= 0.

The average value of the function f(x, y, z) = [tex]3x^2z + 3y^2z[/tex] over the region enclosed by the paraboloid z = 9 −[tex]x^2 - y^2[/tex] and the plane z = 0 can be calculated by integrating the function over the volume and then dividing by the volume. This is somewhat analogous to how one would calculate an average in a discrete distribution.

The volume V(E) of the region E enclosed by the paraboloid and the plane can be found by integrating the equation of the paraboloid over the range of x and y values that satisfy the inequality z = 9 - [tex]x^2 - y^2[/tex] ≥ 0. After finding V(E), you then integrate the function f(x, y, z) over the same range of x, y, and z values to find the total of f over the volume. The average value is then the total divided by V(E).

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According to the Empirical Rule, about 95 percent of the data falls within two standard deviations of the mean. For a basketball player with a mean score of 16 and a standard deviation of 2, the probability of scoring between 12 to 20 points (within 2 standard deviations) in a randomly selected game is approximately 95 percent.

The question here revolves around the concept of the Empirical Rule in the realm of Normal Distribution. The Empirical Rule, which applies to a bell-shaped and symmetrical distribution, states that approximately 68 percent of the data falls within one standard deviation of the mean, 95 percent within two standard deviations, and 99.7 percent within three standard deviations.

In this case, the basketball player's game scores have a mean of 16 and a standard deviation of 2. To find the probability of the player scoring between 12 and 20, we'll use the Empirical Rule. Scores between 12 and 20 are within two standard deviations from the mean (16-4=12 and 16+4=20). Therefore, according to the Empirical Rule, the chance of scoring between these two numbers is about 95 percent.

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Using the empirical rule, the probability that the player scored between 12 and 20 points in a randomly selected game is approximately 95.45%.

To find the probability that the player scored between 12 and 20 points, we can use the empirical rule for a normal distribution. According to the empirical rule, approximately 68% of the data falls within one standard deviation of the mean, and approximately 95% falls within two standard deviations. Since the mean is 16 points and the standard deviation is 2 points, we can calculate the z-scores for 12 and 20 and find the area under the curve between those z-scores.

First, we calculate the z-score for 12: z = (x - μ) / σ = (12 - 16) / 2 = -2. Then, we calculate the z-score for 20: z = (x - μ) / σ = (20 - 16) / 2 = 2. With these z-scores, we can look up the corresponding areas under the standard normal distribution curve in a z-table. The area between -2 and 2 is approximately 0.9545. To find the probability, we subtract the area outside of this range (0.0455) from 1, giving us a probability of approximately 0.9545 or 95.45%.

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Answer:

The Miller Family used the sprinkler for 40 hours.

The Washington Family used the sprinkler for 30 hours.

Step-by-step explanation:

First write an equation.

M = Miller Family's Output Rate

W = Washington Family's Output Rate

30M + 15W = 1650

M + W = 70

Using simultaneous equations:

1) Make one of the coefficients the same value.

We will make both W's 15.

Multiply the second equation by 15.

15M + 15W = 1050

2) Subtract the equations to remove the coefficient.

(30M + 15W = 1650) - (15M + 15W = 1050)

(30M + 15W) - (15M + 15W) = 1650 - 1050

15M = 600

3) Divide to find the value of 1 M

15M = 600

M = 600/15

M = 40

4) Substitute M into either equation to find the value of W.

30M + 15W = 1650

30(40) + 15W = 1650

1200 + 15W = 1650

15W = 1650 - 1200

15W = 450

W = 450/15

W = 30

M + W = 70

40 + W = 70

W = 70 - 40

W = 30

Answer:

Miller family's sprinkler was used for 40 hours and Washington family's sprinkler was used for 30 hours.

Step-by-step explanation:

 Set up a system of equations.

Let be "m" the time Miller family's sprinkler was used and "w" the time Washington family's sprinkler was used.

Then:

[tex]\left \{ {{m+w=70} \atop {30m+15w= 1,650}} \right.[/tex]

You can use the Elimination method. Multiply the first equation by -30, then add both equations and solve for "w":

[tex]\left \{ {{-30m-30w=-2,100} \atop {30m+15w= 1,650}} \right.\\.................................\\-15w=-450\\w=30[/tex]

 Substitute w=30 into an original equation and solve for "m":

[tex]m+30=70\\m=70-30\\m=40[/tex]

Answer:

0.999

Step-by-step explanation:

At least 1 correct means, 1 correct, 2 correct, 3 correct ... until 10 correct. That would be a long process to calculate.

Instead we use the complement rule to calculate.

[tex]P(x\geq1)=1-P(x<1)[/tex]

So we need to find P(x<1). So this is getting 0 answers correct, or 10 incorrect.

In true false question, probablity of correct is 1/2 and incorrect is 1/2, hence,

Probability of 10 incorrect is (1/2)^10

Thus,

[tex]P(x\geq1)=1-(\frac{1}{2})^{10}=0.999[/tex]

So the answer is 0.999 (rounded to nearest thousandth)

An unprepared student's probability of guessing at least one correct answer in a ten-question true-false quiz is 0.999, found by calculating the complement of all answers being incorrect.

To find the probability that an unprepared student makes at least one correct guess on a ten-question true-false quiz, we start by recognizing that each question has two possible answers (true or false), so the probability of guessing correctly on a single question is 0.5.

The complement of guessing at least one correct answer is guessing all answers incorrectly. The probability of guessing one question incorrectly is also 0.5. Therefore, the probability of guessing all ten questions incorrectly is (0.5)¹⁰.

Calculation:

The probability of guessing at least one correct answer is the complement of this probability:

1 - 0.0009765625 = 0.9990234375

Rounding to the nearest thousandth, the probability that the student guesses at least one correct answer is 0.999.

Answer:

The fourth one down

Step-by-step explanation:

In order for the limit of a function to exist, the general limit that is, it has to agree from both the left and the right of the function.  Since this is not asking for a left-handed limit or a right-handed limit, coming in from both the left and right does not equal the same y value.  So the limit does not exist.  

To find f(3), look to where x = 3 and find the y value where the solid dot at x=3 is.  When x = 3, the solid dot has a y value of 5.  Therefore, f(3) = 5.

The answer you want is the fourth one down.

The limit of function at x = 3 doest not exist and value of function will be f(3) = 5 so option (D) will be correct.

Limit, a nearness in mathematical notion, is largely used to give values to some functions at locations that were usually not given or defined, in a manner consistent with neighboring values.

In another word, limit is a mathematical concept in which we find out the value of any function at a point by its adjacent very close point.

Because the value of that function at that point doesn't define.

Given a function f(x)

As we can see that the function is breaking at x = 3 into two functions.

To exist any limit the left-hand limit must be equal to the right-hand limit

Left-hand limit

LHL = 3

Right-hand limit

RHL = 5

Since limits are not equal hence limits do not exist.

Now value of functon f(x) at x = 3 is 5 not 3 becuase at 3 the point is hollow while at x = 5 point is solid.

Hence,the limit of function at x = 3 doest not exist and value of function will be f(3) = 5

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Answer:  New Mean = $40,920

                New Median = $38,500

Step-by-step explanation:

5 employee's salaries are as follows in order from least to greatest:

       $32,000 - unknown - $38,500 - unknown - unknown

The median (middle number) is: $38,500

The mean (average) of the 5 salaries is: $40,300

If $3,100 is added to the $32,000 salary, then the mean(average) is increased by [tex]\dfrac{\$3100}{5\ salaries}=\$620[/tex].

Old Mean + increase = New Mean

$40,300  +  $ 620    = $40,920

The median (middle number) does not change. It is still $38,500

Answer:

The area of the rectangle is 12 cm² ⇒ in 2 significant figures

Step-by-step explanation:

* Lets talk about the significant figures

- All non-zero digits are significant

# 73 has two significant figures

- Zeroes between non-zeros digits are significant

# 105.203 has six significant figures

- Leading zeros are never significant

# 0.00234 has three significant figures

- In a number with a decimal point, zeros to the right of the last

 non-zero digit are significant

# 19.00 has four significant figures

- Lets make a number and then approximate it to different significant

∵ 12.7360 has 6 significant figures

∴ 12.736 ⇒ approximated to 5 significant figures

∴ 12.74 ⇒ approximated to 4 significant figures

∴ 12.7 ⇒ approximated to 3 significant figures

∴ 13 ⇒ approximated to 2 significant figures

∴ 10 ⇒ approximated to 1 significant figure

- Another number with decimal point

∵ 0.0546700 has 6 significant figures

∴ 0.054670 ⇒ approximated to 5 significant figures

∴ 0.05467 ⇒ approximated to 4 significant figures

∴ 0.0547 ⇒ approximated to 3 significant figures

∴ 0.055 ⇒ approximated to 2 significant figures

∴ 0.05 ⇒ approximated to 1 significant figures

* Lets solve the problem

∵ The width of the rectangle is 2.1 cm

∵ The length of the rectangle is 5.6 cm

- Area of the rectangle = length × width

∴ Area of the rectangle = 2.1 × 5.6 = 11.76 cm²

- Approximate it to two significant figures

∴ Area of the rectangle = 12 ⇒ to the nearest 2 significant figures

* The area of the rectangle is 12 cm² ⇒ in 2 significant figures

Answer:

Answer is: 4/3 hrs. or 1 and 1/3 hrs

Time taken by Chris and Larry to mow the lawn is 3/4 hours.

Given that, it took Chris 4 hours to mow the lawn and it took Larry only 2 hours to mow the lawn.

We know that, Time Taken = 1 / Rate of Work

Here, 1/4 + 1/2

= 1/4 + 2/4

= 3/4

Therefore, it took 3/4 hours for Chris and Larry to mow the lawn.

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Answer:

c. 28

Step-by-step explanation:

Congruence means that all the sides and angle of the triangles which are said to be congruent are equal.

In the given triangle, all three angles of GEF and HEF will be equal.

Using the property, we can see that in triangle GEF the angle is 60 degree and in HEF the corresponding congruent angle of it is 2(x+2)

So,

Putting them equal

2(x+2) = 60

Now, it is a simple equation to solve.

2x+4=60

2x+4-4 = 60-4

2x = 56

2x/2 = 56/2

x = 28 degrees

So, option C is the correct answer ..

Answer:

The variable "a number" stands for 9.

Step-by-step explanation:

Rewrite the problem as 2 * (6 + x) = 30

Divide 30 into 2. 30/2 = 15

That means that the variable that is added to 6 must make the number 15.

15 - 6 = 9

The variable x is 9 so the equation would be:

2 * (6 + 9) = 30

Answer:

Step-by-step explanation:

[tex]n-the\ number\\\\\text{two times the sum of 6 and the number}\ n:\ 2(6+n)\\\\\text{The equation:}\\\\2(6+n)=30\qquad\text{divide both sides by 2}\\\\\dfrac{\not2(6+n)}{\not2}=\dfrac{30\!\!\!\!\!\diagup^{15}}{\not2_1}\\\\6+n=15\qquad\text{subtract 6 from both sides}\\\\6-6+n=15-6\\\\n=9[/tex]

Step-by-step explanation:

The energy required to stretch a spring a distance x is:

E = ½ kx²

where k is the stiffness constant.

If spring B has a constant k and is stretched a distance of b:

E = ½ k b²

If spring A has a constant of 4k and is stretched a distance of a:

E = ½ (4k) a²

E = 2 k a²

If the energies are the same:

½ k b² = 2 k a²

b² = 4 a²

b = 2a

So spring B is stretched twice as far as spring A.

When the same amount of energy is applied to both Spring A and Spring B, Spring B which has a lower spring constant stretches more than Spring A. The spring constant has an inverse relationship with the amount of displacement for a given energy.

The problem deals with the physics concept of potential energy stored in springs. When equal amounts of energy are applied to multiple springs, different amounts of displacement or extension will result, depending on the spring constant (k) of each spring.

Based on Hooke's law, the potential energy (U) stored in a spring is given by the equation U = 1/2 kx², where x represents the displacement or extension of the spring.

For Spring A, which has a spring constant four times that of Spring B, while the same amount of energy is applied to both, less displacement will occur compared to Spring B. This is because a spring with a higher spring constant is stiffer and therefore resists displacement more.

So to answer your question, Spring B will stretch more than Spring A when the same amount of energy is applied to both.

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[tex]|\Omega|=5^3=125\\|A|=2\cdot3\cdot1=6\\\\P(A)=\dfrac{6}{125}=\dfrac{48}{1000}=0.048[/tex]

The 6th term of the geometric sequence is:

                  [tex]a_6=-1458[/tex]

We know that the nth term of a geometric sequence is given by the formula:

          [tex]a_n=a_1\cdot r^{n-1}[/tex]

where [tex]a_1[/tex] is the first term of the sequence and r is the common ratio of the sequence and [tex]a_n[/tex] is the nth term of a sequence.

Also,

[tex]a_=-6[/tex]

and [tex]r=3[/tex]

Hence, we get:

[tex]a_6=-6\times (3)^{6-1}\\\\\\a_6=-6\times 3^5\\\\\\a_6=-1458[/tex]

           Hence, the answer is:

                         -1458

Answer:

b. 3

Step-by-step explanation:

Here, n players score 2 points and m players score 5 points,

So, total scores = 2n + 5m,

According to the question,

The total number of points is 50,

⇒ 2n + 5m = 50

Since, n and m can be any positive integers including 0 ( because number of players can not be negative or in fraction )

Also, for the positive integer value of m, n must be the multiple of 5 less than or equal to 25,

Thus, the possible values of m and n are,

(5,8), (10, 6), (15, 4), (25,0),

Since, 8-5 < 10-6 < 15-4 < 25-0

Hence, the least possible positive difference between n and m is 3.

Option 'b' is correct.

Final answer:

The least possible positive difference between the number of players scoring 2 points and 5 points to reach a total of 50 points is 3. This is determined by finding pairs of values (n, m) that satisfy the equation 2n + 5m = 50 and choosing the pair with the smallest difference. The correct option is b.

Explanation:

The question given is a linear Diophantine equation mathematics problem where we need to find the least possible positive difference between the number of players scoring 2 points (n) and the number of players scoring 5 points (m), given that the total points scored is 50.

To solve this, we start with the equation 2n + 5m = 50 and find pairs of values (n, m) that satisfy this equation. We're looking for the pair with the smallest positive difference |n - m|.

Let's look at the possibilities for m and calculate corresponding n values, remembering that both m and n must be non-negative integers:

If m = 0, then 2n = 50, so n = 25 (difference is 25).

If m = 2, then 2n = 40, so n = 20 (difference is 18).

If m = 4, then 2n = 30, so n = 15 (difference is 11).

If m = 6, then 2n = 20, so n = 10 (difference is 4).

If m = 8, then 2n = 10, so n = 5 (difference is 3).

If m = 10, then 2n = 0, so n = 0 (difference is 10, but not a smaller difference).

From these calculations we can see that the smallest positive difference is when m = 8 and n = 5, which is 3. Therefore, the answer is b. 3.1

Answer with explanation:

(a)

Mean number of flight hours for Continental Airline pilots = 49 hours per month

Total Sample Size =100

Standard Deviation =11.5 Hours

Margin of error for a 95% confidence interval

        [tex]=Z_{95 \text{Percent}}\times \frac{\sigma}{\sqrt{n}}\\\\=0.8365 \times \frac{11.5}{\sqrt{100}}\\\\=\frac{9.61975}{10}\\\\=0.961975\\\\=0.97(\text{Approx})[/tex]

(b)

The Range of values for a 95% confidence interval

 ⇒     Mean number of flight  + Margin of Error  ≤  Confidence interval ≤    Mean number of flight  - Margin of Error      

⇒ 49+0.97  ≤  Confidence interval ≤ 49-0.97

⇒ 49.97  ≤  Confidence interval ≤48.03

Upper Bound = 49.97

It will take approximately 154.33 months to pay off a $51,000 loan at a 12% annual interest rate, with monthly payments of $650, using the amortization formula.

To calculate the time it takes to pay off a loan, we can use the formula for the monthly payment in an amortizing loan:

[tex]\[ M = P \times \frac{r(1+r)^n}{(1+r)^n-1} \][/tex]

where:

- M is the monthly payment,

- P is the loan amount,

- r is the monthly interest rate,

- n is the total number of payments.

First, let's calculate the monthly interest rate (r):

[tex]\[ r = \frac{\text{Annual Rate}}{12 \times 100} \][/tex]

For the given problem:

[tex]\[ r = \frac{12\%}{12 \times 100} = 0.01 \][/tex]

Now, we'll use the formula to calculate the total number of payments (n):

[tex]\[ n = \frac{\log\left(\frac{M}{M - Pr}\right)}{\log(1+r)} \][/tex]

For this problem:

[tex]\[ n = \frac{\log\left(\frac{650}{650 - 51000 \times 0.01}\right)}{\log(1 + 0.01)} \][/tex]

Now, let's calculate this:

[tex]\[ n \approx \frac{\log\left(\frac{650}{650 - 510}\right)}{\log(1.01)} \]\[ n \approx \frac{\log\left(\frac{650}{140}\right)}{\log(1.01)} \]\[ n \approx \frac{\log(4.642857)}{\log(1.01)} \]\[ n \approx \frac{0.6654}{0.00432} \]\[ n \approx 154.33 \][/tex]

So, the result is [tex]\( n \approx 154.33 \)[/tex]. Therefore, it will take approximately 154.33 months to pay off the loan.

Answer: 0.2789

Step-by-step explanation:

Given: Mean : [tex]\mu=10.00\ mm [/tex]

Standard deviation : [tex]\sigma =0.03\ mm[/tex]

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

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

For x= 9.96 mm, we have

[tex]z=\dfrac{10-9.96}{0.03}\approx1.33[/tex]

For x= 10.01 mm, we have

[tex]z=\dfrac{10.01-10}{0.03}\approx0.33[/tex]

The P-value = [tex]P(0.33<z<1.33)=P(z<1.33)-P(z<0.33)[/tex]

[tex]= 0.9082408- 0.6293=0.2789408\approx0.2789[/tex]

Hence, the probability that a randomly selected gear has a diameter between 9.96 mm and 10.01 mm =0.2789

Answer:

Pr=0.2894

Step-by-step explanation:

given mean diameter =10 mm

standard deviation=0.03 mm

z equation is

z=x-μ/σ

The problem has two values of x

for x=9.96

z=-1.33

for x-10.01

z=0.33

from Probability table we have

Pr(-1.33<z<0.33)=pr(z<0.33)-pr(z>-1.33)

Pr=0.2894