The average value of a function f(x, y, z) over a solid region E is defined to be fave = 1 V(E) E f(x, y, z) dV where V(E) is the volume of E. For instance, if ρ is a density function, then ρave is the average density of E. Find the average value of the function f(x, y, z) = 3x2z + 3y2z over the region enclosed by the paraboloid z = 9 − x2 − y2 and the plane z = 0.

[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]

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

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: 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:

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]

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: 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:

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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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:

0.999

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 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:

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 ..

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:

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:

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: 0.1353

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

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:  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