what is the measurement of angle p? Round your answer to the nearest degree.


A. 29°

B.42°

C.65°

D.78°

Answer:

  0.252 m

Step-by-step explanation:

At the fundamental frequency, a closed-end column has the same wavelength as an open-end column twice as long. Column B only needs to be half the length of Column A:

  0.504 m/2 = 0.252 m

[tex]a^{p-1} \equiv 1 \pmod p[/tex] where [tex]p[/tex] is prime, [tex]a\in\mathbb{Z}[/tex] and [tex]a[/tex] is not divisible by [tex]p[/tex].

[tex]7^{13-1}\equiv 1 \pmod {13}\\7^{12}\equiv 1 \pmod {13}\\\\542=45\cdot12+2\\\\7^{45\cdot 12}\equiv 1 \pmod {13}\\7^{45\cdot 12+2}\equiv 7^2 \pmod {13}\\7^{542}\equiv 49 \pmod{13}[/tex]

Answer:

49 mod 13 = 10.

Step-by-step explanation:

Fermat's little theorem states that

x^p = x mod p where p is a prime number.

Note that 542 = 41*13 + 9 so

7^542 = 7^(41*13 + 9)  = 7^9 * (7^41))^13

By FLT (7^41)^13 = 7^41 mod 13

So 7^542 = ( 7^9 *  7(41)^13) mod 13

= (7^9 * 7^41) mod 13

= 7^50 mod 13

Now we apply FLT to this:

50 = 3*13 + 11

In a similar method to the above we get

7^50 = (7^11 * (7^3))13)  mod 13

=  (7^11 * 7^3) mod 13

= (7 * 7^13) mod 13

= ( 7* 7) mod 13

= 49 mod 13

= 10 (answer).

Answer:

INF for first while D for second

Step-by-step explanation:

Ok I think I read that integral with lower limit 1 and upper limit infinity

where the integrand is ln(x)*x^2

integrate(ln(x)*x^2)

=x^3/3 *ln(x)- integrate(x^3/3 *1/x)

Let's simplify

=x^3/3 *ln(x)-integrate(x^2/3)

=x^3/3*ln(x)-1/3*x^3/3

=x^3/3* ln(x)-x^3/9+C

Now apply the limits of integration where z goes to infinity

[z^3/3*ln(z)-z^3/9]-[1^3/3*ln(1)-1^3/9]

[z^3/3*ln(z)-z^3/9]- (1/9)

focuse on the part involving z... for now

z^3/9[ 3ln(z)-1]

Both parts are getting positive large for positive large values of z

So the integral diverges to infinity (INF)

By the integral test... the sum also diverges (D)

To compute the value of the improper integral, we can integrate by parts. Using the formula for integration by parts, we find that the integral converges to a finite value of -ln(x)/x as x approaches infinity.

To compute the value of the improper integral ∫∞18ln(x)/x2dx, we can integrate by parts. Let u = ln(x) and dv = 1/x2dx. Differentiating u with respect to x gives du = 1/x dx and integrating dv gives v = -1/x. Applying the formula for integration by parts, we get:

∫∞18ln(x)/x2dx = -ln(x)/x + ∫∞181/x2dx.

Simplifying the integral, we have:

∫∞181/x2dx = -1/x

As x approaches infinity, 1/x approaches 0. Therefore, the improper integral converges to a finite value of -ln(x)/x.

Answer:  (b) Systematic

Step-by-step explanation:

Given statement : To estimate the percentage of defects in a recent manufacturing​ batch, a quality control manager at General Electric selects every 20th refrigerator that comes off the assembly line starting with the sixth until she obtains a sample of 130 refrigerators.

Periodic interval : Every 20th

Beginning point : 6th

Sample size : 130

Hence, this type of sampling is a systematic random sampling.

The type of sampling used by the General Electric quality control manager is systematic sampling. This method involves selecting subjects at fixed regular intervals, providing a balance between bias and needed sample size.

The quality control manager at General Electric is using systematic sampling to estimate the percentage of defects in a recent manufacturing batch. Systematic sampling involves selecting subjects at regular intervals from the population, in this case, every 20th refrigerator starting with the sixth one to achieve a sample of 130 refrigerators. This type of sampling is situated between other sampling methods such as random sampling and judgmental sampling in terms of bias and sample size needed to characterize the population adequately.

When considering the provided exercises, the answers to the type of sampling used would be:

Answer:

Step-by-step explanation:

The combined price of the separate obligations is $2000, so the package price is 1900/2000 = 0.95 of the total of separate items. We assume the allocation matches that proportion, so the allocations are ...

  TV: 0.95×$1500 = $1425

  remote: 0.95×$200 = $190

  installation: 0.95×$300 = $285

The revenue that would be allocated to the TV, the remote, and the installation service are;

New revenue for TV = $1425

New revenue for remote = $190

New revenue for installation= $285

Cost of 60-inch TV = $1,500

Cost of remote = $200

Installation service cost = $300

Total revenue to be generated when they pay individually = 1500 + 200 + 300 = $2000

This means, the discount here is; 1900/2000 × 100 = 95% or 0.95

New revenue for TV: 0.95 × $1500 = $1425

New revenue for remote: 0.95 × $200 = $190

New revenue for installation: 0.95 × $300 = $285

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Answer: Probability that both are non smokers is 0.48.

Step-by-step explanation:

Since we have given that

Number of male smokers = 102

Number of male non smokers = 408

Total male = 102+408=510

Number of female smokers = 240

Number of female non smokers = 360

Total female = 240+360 = 600

According to question,

A male student and a female student from the college are randomly selected for a survey,

So, Probability that both are non smokers is given by

P(both are smokers ) = P(Male smoker) × P(female smoker)

[tex]P(both)=\dfrac{360}{600}\times \dfrac{408}{510}\\\\P(both)=\dfrac{146880}{306000}\\\\P(both)=0.48[/tex]

Hence, probability that both are non smokers is 0.48.

The probability that both the randomly selected male and female students from the college are non-smokers is 0.48.

To compute the probability that both the randomly selected male and female students are non-smokers, we start by finding the total number of male and female students. The total number of male students is 510 (102 smokers + 408 non-smokers), and the total number of female students is 600 (240 smokers + 360 non-smokers). Then, we calculate the individual probabilities of selecting a non-smoking male and a non-smoking female. The probability of selecting a non-smoking male is 408 / 510, and the probability of selecting a non-smoking female is 360 / 600.

To find the combined probability, we multiply these individual probabilities. So, the probability that both the male and the female students selected are non-smokers, is (408 / 510) * (360 / 600) = 0.8 * 0.6 = 0.48.

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(a) The probability of answering all 5 problems correctly is 0.08.

(b) The probability of answering at least 4 problems correctly is 0.5.

There are a total of 10 problems, and the student has figured out how to solve 7 of them which means there are 3 problems that the student hasn't figured out how to solve.

(a) To find the probability that the student answers correctly to all 5 problems, we need to consider that the student must select 5 out of the 7 problems they know how to solve and 0 out of the 3 problems they don't know how to solve.

The probability of selecting 5 specific problems out of 7 is given by:

(7 choose 5) / (10 choose 5)  [tex]=\frac{^7C_5}{^{10}C_5}[/tex]

=21/252

=0.083

(b) To find the probability that the student answers at least 4 problems correctly, we need to consider two cases: when the student answers 4 problems correctly and when the student answers all 5 problems correctly.

Case 1: Student answers 4 problems correctly and 1 problem incorrectly:

P(4 correct and 1 incorrect) = (7 choose 4) × (3 choose 1) / (10 choose 5)

Case 2: Student answers all 5 problems correctly:

P(5 correct out of 5) = (7 choose 5) / (10 choose 5)

Now, add the probabilities of these two cases to get the probability of answering at least 4 problems correctly:

P(at least 4 correct) = P(4 correct and 1 incorrect) + P(5 correct out of 5)

Calculate each part using combinations:

P(4 correct and 1 incorrect) = (35 × 3) / 252

= 0.4167

P(5 correct out of 5) = 0.08 (as calculated in part a)

Now, add these probabilities:

P(at least 4 correct) = 0.4167 + 0.08

= 0.5

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The probability that the student will correctly answer all 5 questions is 0.083 (8.3%) and the probability of correctly answering at least 4 questions is 0.5 (50%).

This is a probability question related to the field of combinatorics. To answer the student's question, we have to calculate the probability of correctly answering the questions out of the known ones.

The total ways the instructor can select 5 problems out of 10 is represented by the combination formula C(10,5). This equates to 252.

The student can answer 7 questions, so a) the number of ways of getting all 5 correctly is represented by C(7,5) which equals 21. Therefore, the probability of answering all 5 correctly is 21/252 = 0.083 or 8.3%.

For b), the student aims to answer at least 4 correctly. This means that we calculate the probability for getting 4 and 5 problems correct. For 4 problems it's C(7,4)*C(3,1) = 105. Adding the ways to get 5 problems correct, we get 105+21=126. So, the chance of answering at least 4 correct is 126/252 = 0.5 or 50%.

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We know that in a frequency table we write the frequency corresponding to each of the data.

The relative frequency is the ratio of the frequency to the total frequency corresponding to each entry.

Here we have a total sample as: 120

Also, the frequency corresponding to A is: 60

Corresponding to B is: 23

and corresponding to C is: 37

Hence, the Frequency table is as follows:

Data               Frequency

 A                          60  

 B                           23

 C                           37

The relative frequency table is given by:

Data               Relative Frequency          

 A                      60/120=0.5

 B                       23/120=0.19  

 C                       37/120=0.31

Hence, we get:

Data               Relative Frequency          

 A                           0.5

 B                           0.19

 C                           0.31

The frequency distribution for the responses A, B, and C are 60, 23, and 37 respectively. The relative frequency distribution for the responses A, B, and C are 0.50, 0.19, and 0.31 respectively. Sum of the relative frequencies equals to 1, indicating all portions of the sample have been accounted for.

To create these distributions for the given response data, we'll organize our observations into a table.

Frequency distribution: The frequency shows the number of times each response (A, B, or C) occurs in the sample.
- Response A: 60
- Response B: 23
- Response C: 37

Adding these frequencies would equal the total sample size, 120.

Relative frequency distribution: The relative frequency is calculated as the frequency of each response divided by the total sample size, expressed as a decimal.
- Response A: 60/120 = 0.50
- Response B: 23/120 = 0.19
- Response C: 37/120 = 0.31

Note that if we add up the relative frequencies, we should get a sum of 1, indicating that we've accounted for all portions of the sample.

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Answer:

[tex]-\frac{13\sqrt{x} }{x}[/tex]

Step-by-step explanation:

We have been given the following expression;

-13/√x

In order to rationalize the denominator, we multiply the numerator and the denominator by √x;

[tex]-\frac{13}{\sqrt{x}}=-\frac{13\sqrt{x} }{\sqrt{x}\sqrt{x}}\\ \\-\frac{13\sqrt{x} }{\sqrt{x^{2} } }\\\\-\frac{13\sqrt{x} }{x}[/tex]

Final answer:

To rationalize the denominator, multiply both the numerator and denominator by the conjugate of the denominator. In this case, the conjugate of √x is -√x. Multiply -13/√x by -√x/-√x to get -13√x / x.

Explanation:

To rationalize the denominator, we need to eliminate the square root from the denominator. We can do this by multiplying the numerator and denominator by the conjugate of the denominator. The conjugate of √x is -√x. So, multiplying the numerator and denominator by -√x gives us:

-13/√x * (-√x)/(-√x) = 13√x/(-x) = -13√x / x

Therefore, the rationalized form of -13/√x is -13√x / x.

Answer:

A.) 3/7

B.) 5/7

C.) 2/7

D.) 2/7

Answer:

-3.3

Step-by-step explanation:

-2.3-(4.5-3 1/2)=-2.3-(4.5-3.5)=-2.3-(1)=-3.3

Answer:

less than 3

Step-by-step explanation:

[tex]|\Omega|={_{48}C_6}=\dfrac{48!}{6!42!}=\dfrac{43\cdot44\cdot45\cdot46\cdot47\cdot48}{720}=12271512\\|A|={_6C_4}=\dfrac{6!}{4!2!}=\dfrac{5\cdot6}{2}=15\\\\P(A)=\dfrac{15}{12271512}=\dfrac{5}{4090504}\approx1.2\%[/tex]

The probability that the player wins the prize from a drawn of six in 1 to 48 numbers is 0.001052.

Probability is usually expressed as the number of favorable outcomes divided by the number of desired outcomes.

Mathematically;

Probability of winning = [tex]\mathbf{\dfrac{favorable \ outcomes}{number \ of \ desired \ outcomes}}[/tex]

The probability that the player wins can be computed by taking the following combinations.

From the given information;

Thus, the probability of winning can now be computed as:

[tex]\mathbf{P(winning \ prize) = \dfrac{^6C_4 \times ^{42}C_2}{^{48}C_6}}[/tex]

[tex]\mathbf{P(winning \ prize) = \dfrac{\Big (\dfrac{6!}{4!(6-4)!} \Big )\times \Big (\dfrac{42!}{2!(42-2)!} \Big )}{\Big (\dfrac{48!}{6!(48-6)!} \Big )}}[/tex]

[tex]\mathbf{P(winning \ prize) = \dfrac{12915}{12271512}}[/tex]

[tex]\mathbf{P(winning \ prize) =0.001052}[/tex]

Therefore, the probability that the player wins the prize from a drawn of six in 1 to 48 numbers is 0.001052.

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Answer:

y = 85x + 55

Step-by-step explanation:

If we use the given prices as coordinates we will get the coordinates (2, 225) and (5, 480) and if we use the rise over run which in this case is 255/3 then we get the slope which is $85 and now we plug what we know into y = mx + b. we now have the equation 225 = (85)2 + b which can be simplified to 55 = b and now we can create the equation y = 85x + 55.

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Final answer:

The twenty-fifth term of the arithmetic sequence is 119. The first three terms of another arithmetic sequence are 17, 19, 21.

Explanation:

To find the twenty-fifth term of an arithmetic sequence, we can use the formula:
nth term = first term + (n-1) * common difference

Substituting the given values:
nth term = -1 + (25-1) * 5 = -1 + 24 * 5 = -1 + 120 = 119

Therefore, the twenty-fifth term of the arithmetic sequence is 119.

For the second part of the question, to find the common difference, we can use the formula:
common difference = (fiftieth term - twenty-first term) / (50 - 21)

Substituting the given values:
common difference = (75 - 17) / (50 - 21) = 58 / 29 = 2

Using the first term of 17 and the common difference of 2, we can write the first three terms of the arithmetic sequence:
17, 19, 21

Answer:

When you find the gradient (slope) of a graph, you divide a change of value on the vertical-axis (the 'rise') by a change of value on the horizontal axis (the 'run').  

Gradient = rise/run.  

The vertical axis has units of cm, so the rise in in cm.  

The horizontal axis has units of 1/grams = g⁻¹, so the rise is in g⁻¹.  

units for slip are  

rise/run ≡ cm/g⁻¹ ≡ cm.g

Step-by-step explanation:

The slope in the plotted graph in Excel is determined by the change in the vertical value ('x' - distance, in centimeters) with the change in the horizontal value ('1/m' - inverse of mass, in grams). Therefore, the unit of the slope will be grams centimeters (g.cm).

In this lab experiment using Excel to plot a graph, it's vital to understand the concept of slope. In a line graph, slope indicates the amount of vertical 'rise' for every unit of horizontal 'increase' and is calculated by the change in the dependent variable (in this case 'x' or distance, which is on the vertical axis) over the change in the independent variable (here, it is 1/mass or 1/m, on the horizontal axis).

Since 'x' is measured in centimeters (cm) and 'm', the mass, is measured in grams (g), when calculating the slope, we do a division of cm by the inverse of grams (1/g), which is equivalent to multiplication by its reciprocal (g/1). Therefore, the unit of the slope of the line would be g.cm, which is grams centimeters.

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The equation of the tangent plane to the ellipsoid at the given point is [tex]\frac{xx^0}{a^2} + \frac{yy^0}{b^2} - \frac{zz^0}{c^2} = 1[/tex]

The equation of the tangent plane to the ellipsoid is given as:

[tex]\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1[/tex]

The point on the ellipsoid is given as:

(x0, y0, z0)

The equation of the tangent plane to the ellipsoid at the given point can be written as:

[tex]\frac{xx^0}{a^2} + \frac{yy^0}{b^2} + \frac{zz^0}{c^2} = 1[/tex]

Given that the equation of the tangent plane to the hyperboloid is

[tex]\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1[/tex]

The equations at the tangents of the ellipsoid and the hyperboloid take the same form.

So, the equation of the tangent plane to the ellipsoid at the given point is [tex]\frac{xx^0}{a^2} + \frac{yy^0}{b^2} - \frac{zz^0}{c^2} = 1[/tex]

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To calculate the 95% confidence interval for the mean life span of a sample of 105 people, use the formula Confidence interval = sample mean ± (z-score)*(standard deviation/√n). The z-score for a 95% confidence level is approximately 1.96.

The subject of this question is regarding statistics, specifically the calculation of confidence intervals for a sample mean. In this case, we will use the formula for a confidence interval for the mean:

Confidence interval = sample mean ± (z-score)*(standard deviation/√n)

Where the sample mean is the mean life span of humans (89.87 years), n is the sample size (105 people), the standard deviation is 16.63 years and the z-score corresponds with 95% confidence level (approximately 1.96). After performing the necessary computations:

Confidence interval = 89.87 ± (1.96 * 16.63/√105)

The final step would be calculating the upper and lower bounds by adding and subtracting the product from the mean respectively. Your results will indicate the range within which 95% of sample means of life spans at birth are expected to fall.

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Answer: 4.29203

Explanation:

5^x=1008-8

5^x=1000

take the log of both sides

x=3 log 5 (10)

x=3+3log5(2)

or 4.29203

For this case we must solve the following equation:

[tex]8 + 5 ^ x = 1008[/tex]

Subtracting 8 on both sides of the equation:

[tex]5 ^ x = 1008-8\\5 ^ x = 1000[/tex]

We apply Neperian logarithm to both sides of the equation:

[tex]ln (5 ^ x) = ln (1000)[/tex]

We use the rules of the logarithms to draw x

of the exponent.

[tex]xln (5) = ln (1000)[/tex]

We divide both sides of the equation between[tex]ln (5)[/tex]:

[tex]x = \frac {ln (1000)} {ln (5)}\\x = 4.29202967[/tex]

Rounding:

[tex]x = 4.2920[/tex]

Answer:

[tex]x = 4.2920[/tex]

Answer:

  (x, y, z) = (1, 1/3, 3 1/3)

Step-by-step explanation:

The normal to plane ABC can be found as the cross product ...

  AB×BC = (1, 0, 2)×(2, -2, -3) = (4, 1, 2)

Then the equation of the plane is ...

  4x +y +2z = 4·0 +5 +2·3 . . . . using point C to find the constant

  4x +y +2z = 11

__

The direction vector of the reference line is the vector of coefficients of t: (0, -1, 2). Then the line through point Q is ...

  (x, y, z) = (1, 2, 0) +t(0, -1, 2) = (1, 2-t, 2t)

__

The value of t that puts a point on this line in plane ABC can be found by substituting these values for x, y, and z in the plane's equation.

  4(1) +(2 -t) +2(2t) = 11

Solving for t gives ...

  t = 5/3

so the point of intersection of the plane and the line is

  (x, y, z) = (1, 2-t, 2t) = (1, 2-5/3, 2·5/3) = (1, 1/3, 3 1/3)

Answer:

Question 6

There is not enough information to determine congruency

Question 9

yes; The triangles are congruent by hypotenuse-leg congruence

Step-by-step explanation:

* Lets revise the cases of congruence  

- SSS  ⇒ 3 sides in the 1st Δ ≅ 3 sides in the 2nd Δ

- SAS ⇒ 2 sides and including angle in the 1st Δ ≅ 2 sides and  

 including angle in the 2nd Δ  

- ASA ⇒ 2 angles and the side whose joining them in the 1st Δ  

 ≅ 2 angles and the side whose joining them in the 2nd Δ

- AAS ⇒ 2 angles and one side in the first triangle ≅ 2 angles  

 and one side in the 2ndΔ

- HL ⇒ hypotenuse - leg of the first right angle triangle ≅ hypotenuse

 - leg of the 2nd right angle Δ

- HA ⇒ hypotenuse - angle of the first right angle triangle ≅ hypotenuse

 - angle of the 2nd right angle Δ

- LL ⇒ leg - leg of the first right angle triangle ≅ leg - leg of the

 2nd right angle Δ

* Now lets solve the problem

# Question 6

- There are two right angle triangles

- They have two different legs

- There is no mention abut legs equal each other

- There is no mention about hypotenuses equal each other

- There is no mention about acute angels equal each other

∴ There is not enough information to determine congruency

# Question 9

- There are two right angle triangles

- Their hypotenuse is common

- There are two corresponding legs are equal

∵ The hypotenuse is a common in the two right triangles

∵ Two corresponding legs are equal

- By using hypotenuse-leg congruence

∴ yes; The triangles are congruent by hypotenuse-leg congruence

Final answer:

There are 10 different radar detectors one can buy, considering the power source, detection capabilities (radar, laser, or both), and model type (no-frills or fancy).

Explanation:

To find out how many different radar detectors one can buy, we need to consider the options presented and calculate the total number of combinations. According to the problem statement, radar detectors are powered either by their own battery or by plugging into the cigarette lighter socket. They come in two models: no-frills and fancy. Moreover, battery-powered detectors can detect either radar, laser, or both, while plug-in types can only detect either radar or laser, but not both.

Adding these up, we get a total of 6 options for battery-powered and 4 options for plug-in detectors, making a grand total of 10 different radar detectors one can buy.

To determine who has the weight that is more extreme relative to their group, we need to calculate the z-scores for both the male and female who weigh 1700 g.

To determine who has the weight that is more extreme relative to their group, we need to calculate the z-scores for both the male and female who weigh 1700 g.

The z-score formula is: z = (x - mean) / standard deviation.

For the male who weighs 1700 g:
z = (1700 - 3277.9) / 571.6 = -1.956.

For the female who weighs 1700 g:
z = (1700 - 3091.6) / 625.7 = -1.394.

Since the absolute value of -1.956 is greater than the absolute value of -1.394, the male with a weight of 1700 g is more extreme relative to their group.

Answer:

  c.  1,464,843

Step-by-step explanation:

The sum of n terms of a geometric sequence with first term a1 and common ratio r is given by ...

  sn = a1(r^n -1)/(r -1)

Filling in the values a1=3, r=5, n=9, we get ...

  s9 = 3(5^9 -1)/(5 -1) = 1,464,843

Answer: -976,563 PLEASE READ DESCRIPTION

Step-by-step explanation:

My question had the same exact numbers but some of them were negative! Please be sure that you have the exact same numbers as me before putting my answer!!

"Find the sum of the geometric sequence −3, 15, −75, 375, ... when there are 9 terms and select the correct answer below"

-3 X -5 = 15 X -5 = -75 X -5 = 375.

Ratio = -5

Use the formula [tex]s_{n} = \frac{a_{1 - a_1 (r)^{n} }}{1 - r}[/tex].

Our [tex]a_1[/tex] (first term) = -3, r (ratio)= -5, and n (number of terms) = 9. Knowing this, plug them into the equation.

[tex]s_9 = \frac{-3 - (-3)(-5)^9}{1-(-5)}[/tex].

First, simplify the exponent. -5 to the ninth power = -1,953,125. Multiply this by the nearest number in exponents (-3). -1,953,125 X -3 = 5,859,375. Continue simplifying your numerator. -3 - (5,859,375) = -5,859,378. Now, simplify your denominator. 1 - (-5) = 6.

Divide.

[tex]s_9 = \frac{-5,859,372}{6}[/tex] = -976,563

I tried to make sure there weren't any typos, but please comment if there's something wrong!

The answer is D. Gym B and Gym A charge the same hourly rate for childcare.

Answer:

  Gym B and Gym A charge the same hourly rate for childcare

Step-by-step explanation:

The rate of $0.75 per 5 minutes is ...

  $0.75/(5/60 h) = $0.75×12/h = $9.00/h

Gym B's hourly rate is $9.00, the same as that of Gym A.

Answer: 0.030

Step-by-step explanation:

Given: The percent of the items produced by a machine are defective  :[tex]P=10\%=0.10[/tex]

The percent of the items produced by machine which are not defective:[tex]Q=1-0.1=0.9[/tex]

Sample size : n = 100

Now, the standard error of the proportion is given by :-

[tex]\text{S.E.}=\sqrt{\dfrac{0.1\times0.9}{100}}\\\\\Rightarrow\text{S.E.}=0.030[/tex]

Hence, the the standard error of the distribution of the sample proportions=0.030

The standard error for the distribution of sample proportions of a machine producing 10% defective items determined from a random sample of 100 items is ±0.030.

Ten percent of the items produced by a machine are defective. From a random sample of 100 items, the standard error of the sample proportions is calculated using the formula for the standard error of a proportion, which is √(pq/n), where p is the proportion of successes, q is the proportion of failures (1-p), and n is the sample size.

In this case, p is 0.10 (the proportion defective), q is 0.90 (1-p, representing the proportion not defective), and n is 100 (the sample size). Substituting these values into the formula gives: √[(0.10)(0.90) / 100] which equals ±0.030 (rounded to three decimal places).

Therefore, the standard error for the distribution of sample proportions is ±0.030.

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The volume of the solid is (81π - 128)/3 cubic units, found by integrating the volume of washer-shaped slices.

Sure, here is the step-by-step calculation of the volume V of the solid formed by rotating the region bounded by the curves y = −[tex]x^2[/tex] + 10x − 16, y = 0, about the x-axis:

Find the x-intercepts.

Set the two equations equal to each other to find the x-coordinates of the points of intersection:

[tex]-x^2 + 10x -16 = 0[/tex]

Factor the expression:

(x - 2)(x - 8) = 0

Therefore, the x-intercepts are x = 2 and x = 8.

Sketch the curves and the axis of rotation.

On a coordinate plane, graph the parabola y = −[tex]x^2[/tex] + 10x − 16 and the line y = 0. The x-axis is the axis of rotation.

Imagine the solid of revolution.

When the shaded region between the parabola and the x-axis is rotated about the x-axis, it forms a solid of revolution. This solid can be imagined as a collection of thin slices.

Consider one such slice.

Take a thin slice of the solid perpendicular to the x-axis. The slice is a cylinder with a hole in the middle, much like a washer. Let the thickness of the slice be dx and let the radius of the washer, as a function of x, be r(x).

Express the volume of the slice as a washer.

The volume of the washer is the difference between the volume of the larger cylinder and the volume of the smaller cylinder inside it. The volume of a cylinder is π[tex]r^2[/tex]h, where r is the radius and h is the height. In this case, the height of each cylinder is dx.

Therefore, the volume of the washer is:

π[([tex]r(x))^2 - (r'(x))^2[/tex])] dx

Express the radius as a function of x.

The radius of the washer is equal to the distance between the parabola and the x-axis. In other words, for any x-value, r(x) = −[tex]x^2[/tex] + 10x − 16.

Set up the definite integral.

To find the total volume of the solid, we need to sum the volumes of infinitely many such washers as the thickness dx approaches zero. This is done using a definite integral:

[tex]\int _a^b \pi[(r(x))^2 - (r'(x))^2] dx[/tex]

where a and b are the x-coordinates of the endpoints of the region. In this case, a = 2 and b = 8.

Differentiate r(x) to find r'(x).

r'(x) = -2x + 10

Evaluate the definite integral.

[tex]\int_2^8 \pi[(-x^2 + 10x - 16)^2 - (-2x + 10)^2] dx[/tex]

This integral can be evaluated using integration by parts or a computer algebra system. The result is:

(81π - 128)/3

Therefore, the volume V of the solid is (81π - 128)/3 cubic units.

Question:

The region bounded by the given curves is rotated about the specified axis. Find the volume V of the resulting solid by any method. y = −[tex]x^2[/tex] + 10x − 16,    y =0;  about the x-axis.

Answer:

42

Step-by-step explanation:

Note that as x increases by 2 from 2 to 4, y increases by 12 from 12 to 24.  Thus, the slope of the line connecting the given points is m = rise / run = 12/2, or m = 6.  Thus, this direct proportion is written as y = 6x.

If x = 7, y = 6(7) = 42

Answer:

42

Step-by-step explanation:

Answer:

The sequence is,

-6, -18, -54, -162, - 486, -1458, -4374

Step-by-step explanation:

It is given a geometric sequence,

–6, ? , ? , ? , ? , –1,458?

From the given sequence we get first term a₁ = -6 and 6th term a₆ = -1458

To find the common ratio 'r'

6th term can be written as

a₆ = ar⁽⁶ ⁻ ¹⁾

-1458 = 6 * r⁽⁶⁻¹⁾

r⁵ = -1458/-6 = 243

r = ⁵√243 = 3

To find the sequence

We have a = -6, r = 3

a₂ = -6 * 3 = -18

a₃ = a₂*3 = -18* 3 = -54

a₄  = a₃*3 = -54 * 3 = -162

a₅ = a₄*3 = -162* 3 = -486

a₆ = - 1458

a₇ = a₂*3 =-1458 * 3= -4374

The sequence is,

-6, -18, -54, -162, - 486, -1458, -4374

Let [tex]x[/tex] be the amount (in mL) of the first brand (7% vinegar) and [tex]y[/tex] the amount of the second brand (15% vinegar). She wants to end up with a mixture with volume 240 mL, so that

[tex]x+y=240[/tex]

and she wants it contain 13% vinegar. Each mL of the first brand contributes 0.07 mL (i.e. 7% of 1 mL) vinegar, while each mL of the second brand contributes 0.15 mL (i.e. 15% of 1 mL). The final mixture needs to contribute 0.13 mL (i.e. 13% of 1 mL) for each mL of dressing, so that

[tex]0.07x+0.15y=0.13(x+y)=31.2[/tex]

Now solve:

[tex]x+y=240\implies y=240-x[/tex]

[tex]0.07x+0.15y=31.2\implies0.07x+0.15(240-x)=31.2[/tex]

[tex]\implies-0.08x+36=31.2[/tex]

[tex]\implies4.8=0.08x[/tex]

[tex]\implies\boxed{x=60}[/tex]

[tex]y=240-x\implies\boxed{y=180}[/tex]

The chef needs to use 60 mL of the first brand and 180 mL of the second brand.

Answer:

First brand: 60 milliliters

Second brand: 180 milliliters

Step-by-step explanation:

Let's call m the amount of the first dressing mark that contains 7% vinegar

Let's call n the amount of the second dressing mark that contains 15% vinegar

The resulting mixture should have 13% vinegar and 240 milliliters.

Then we know that the total amount of mixture will be:

[tex]m + n = 240[/tex]

Then the total amount of vinegar in the mixture will be:

[tex]0.07m + 0.15n = 0.13 * 240[/tex]

[tex]0.07m + 0.15n = 31.2[/tex]

Then we have two equations and two unknowns so we solve the system of equations. Multiply the first equation by -0.15 and add it to the second equation:

[tex]-0.15m -0.15n = 240 * (- 0.15)[/tex]

[tex]-0.15m -0.15n = -36[/tex]

[tex]-0.15m -0.15n = -36[/tex]

              +

[tex]0.07m + 0.15n = 31.2[/tex]

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

[tex]-0.08m = -4.8[/tex]

[tex]m = \frac{-4.8}{-0.08}[/tex]

[tex]m = 60\ milliliters[/tex]

We substitute the value of m into one of the two equations and solve for n.

[tex]m + n = 240[/tex]

[tex]60 + n = 240[/tex]

[tex]n = 180\ milliliters[/tex]