A large school district in southern California asked all of its eighth-graders to measure the length of their right foot at the beginning of the school year, as part of a science project. The data show that foot length is approximately Normally distributed, with a mean of 23.4 cm and a standard deviation of 1.7 cm. Suppose that 25 eighth-graders from this population are randomly selected. Approximately what is probability that the sample mean foot length is less than 23 cm?

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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Answer:

we need to prove : for every integer n>1, the number [tex]n^{5}-n[/tex] is a multiple of 5.

1) check divisibility for n=1, [tex]f(1)=(1)^{5}-1=0[/tex]  (divisible)

2) Assume that [tex]f(k)[/tex] is divisible by 5, [tex]f(k)=(k)^{5}-k[/tex]

3) Induction,

[tex]f(k+1)=(k+1)^{5}-(k+1)[/tex]

[tex]=(k^{5}+5k^{4}+10k^{3}+10k^{2}+5k+1)-k-1[/tex]

[tex]=k^{5}+5k^{4}+10k^{3}+10k^{2}+4k[/tex]

Now, [tex]f(k+1)-f(k)[/tex]

[tex]f(k+1)-f(k)=k^{5}+5k^{4}+10k^{3}+10k^{2}+4k-(k^{5}-k)[/tex]

[tex]f(k+1)-f(k)=k^{5}+5k^{4}+10k^{3}+10k^{2}+4k-k^{5}+k[/tex]

[tex]f(k+1)-f(k)=5k^{4}+10k^{3}+10k^{2}+5k[/tex]

Take out the common factor,

[tex]f(k+1)-f(k)=5(k^{4}+2k^{3}+2k^{2}+k)[/tex]      (divisible by 5)

add both the sides by f(k)

[tex]f(k+1)=f(k)+5(k^{4}+2k^{3}+2k^{2}+k)[/tex]

We have proved that difference between [tex]f(k+1)[/tex] and [tex]f(k)[/tex] is divisible by 5.

so, our assumption in step 2 is correct.

Since [tex]f(k)[/tex] is divisible by 5, then [tex]f(k+1)[/tex] must be divisible by 5 since we are taking the sum of 2 terms that are divisible by 5.

Therefore, for every integer n>1, the number [tex]n^{5}-n[/tex] is a multiple of 5.

Final answer:

The statement is true for all integers greater than 1.

Explanation:

The n5 - n is a multiple of 5 for every integer n > 1, we will use proof by induction.

Base Case: For n = 2,
n5 - n = 25 - 2 = 32 - 2 = 30,
which is clearly a multiple of 5. Hence, our base case holds true.

Inductive Step: Assume that for some integer k > 1, the statement holds true, i.e.,
k5 - k is a multiple of 5. We need to show that k5 - k + 5(k4 + k3 + k2 + k + 1) is also a multiple of 5.

If k5 - k is a multiple of 5, then adding a number which is a multiple of 5 (5 times a sum of powers of k) to it will also result in a multiple of 5. This means that (k + 1)5 - (k + 1) will be a multiple of 5, hence the statement holds for k + 1. By the principle of mathematical induction, the statement holds true for all integers n > 1.

[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: 11

Step-by-step explanation:

5% × 220 =

(5 ÷ 100) × 220 =

(5 × 220) ÷ 100 =

1,100 ÷ 100 =

11;

5% of 220 = 11

Answer:

44% is your answer.

Step-by-step explanation:

So, 220/5= 44/1= 44%

Because if you divide 220 by 5 you'll get 44 and 5 divided by 5 ofcourse is 1.

(I'm doing these also in my class, so hopefully i helped you.)

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.

The gradient of function f is (∂f/∂x, ∂f/∂y) = (2cos(2x + 5y), 5cos(2x + 5y)). The gradient at the point P(-5, 2), is ∇f(-5, 2) = (2cos(-20), 5cos(-20)). Rate of change of f at P in the direction of the vector u is  (-1/2, 3). Duf(-5, 2) = ∇f(-5, 2) · (-1/2, 3).

(a) To find the gradient of the function f(x, y) = sin(2x + 5y), we need to compute the partial derivatives with respect to x and y:

Gradient of f(x, y) = (∂f/∂x, ∂f/∂y).

Taking the partial derivative with respect to x:

∂f/∂x = ∂/∂x(sin(2x + 5y)) = 2cos(2x + 5y).

Taking the partial derivative with respect to y:

∂f/∂y = ∂/∂y(sin(2x + 5y)) = 5cos(2x + 5y).

So, the gradient of f is (∂f/∂x, ∂f/∂y) = (2cos(2x + 5y), 5cos(2x + 5y)).

(b) To evaluate the gradient at point P(-5, 2), we substitute these values into the gradient expression:

∇f(-5, 2) = (2cos(2(-5) + 5(2)), 5cos(2(-5) + 5(2))).

Calculating these values gives the gradient at P.

The gradient at point P is ∇f(-5, 2) = (2cos(-20), 5cos(-20)).

(c) To find the rate of change of f at point P(-5, 2) in the direction of the vector u = (1/2, 3) - (1, 0) = (-1/2, 3), we use the dot product:

Duf(-5, 2) = ∇f(-5, 2) · (-1/2, 3).

Calculate this dot product to find the rate of change of f in the direction of u at point P.

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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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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.

Hope this helps and please mark brainliest :)

Answer:

The second choice is correct.

Step-by-step explanation:

    These are triangles, so the interior angles have to add up to 180 degrees. Since 45 degrees and m<1 are vertical angles, they will have the same measure. So right off the back we know that m<1 =45. To find the measure<2 all you need to do is add 45 and 59, then subtract that answer from 180. M<2= 76. That leaves only one option. The second  one.

The measurements of the angles are Angle 1 = 45 degrees, Angle 2 = 76 degrees, Angle 3 = 80 degrees. The correct option is b) 1 = 45, 2 = 76, 3 = 80.

Let's solve this step by step to find the measurements of angles 1, 2, and 3.

We named the two triangles as OAB and OCD, with a common vertex O. angle O makes vertically opposite angle 45 degree in triangle OAB and angle 1 in OCD, angle A makes angle 2, angle B makes angle 59 degree, angle C makes 55 degree and angle D makes angle 3.Given the information:

In triangle OAB:

Angle OAB (angle 2) = 59 degrees

Angle O = 45 degrees (vertically opposite to angle 1)

In triangle OCD:

Angle O = 45 degrees (again, vertically opposite to angle 1)

Angle OCD (angle 3) = ?

First, let's find angle 2 in triangle OAB:

In triangle OAB, the sum of angles in a triangle is 180 degrees.

So, angle 2 + angle O + angle OAB = 180 degrees.

59 + 45 + angle OAB = 180 degrees.

Now, solve for angle OAB:

angle OAB = 180 degrees - 59 degrees - 45 degrees

angle OAB = 76 degrees

Now, let's find angle 3 in triangle OCD:

In triangle OCD, the sum of angles in a triangle is also 180 degrees.

So, angle O + angle OCD + angle C = 180 degrees.

45 + angle 3 + 55 = 180 degrees.

Now, solve for angle 3:

angle 3 = 180 degrees - 45 degrees - 55 degrees

angle 3 = 80 degrees

So, the measurements of angles 1, 2, and 3 are as follows:

Angle 1 = 45 degrees

Angle 2 = 76 degrees

Angle 3 = 80 degrees

The correct option is:

b) 1 = 45, 2 = 76, 3 = 80

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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!

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)

To find the probability of the soft drink machine filling a cup between 30 and 31 ounces, calculate the z scores for 30 and 31 ounces, use them to find the cumulative probabilities from a standard normal distribution table, then subtract the two probabilities. The result is 0.0919 or 9.19%.

This is a question about probability in a normal distribution. In this case, we want to find the probability of the output being between 30 and 31 ounces, given a mean of 28 ounces and a standard deviation of 2 ounces.

First, we find the z-scores for 30 and 31 ounces. The z-score is calculated by subtracting the mean from the value and dividing the result by the standard deviation. For 30 ounces, the z-score is (30-28)/2 = 1. For 31 ounces, the z-score is (31-28)/2 = 1.5.

Next, we use these z-scores to find the cumulative probabilities from a standard normal distribution table. The cumulative probability for a z-score of 1 is 0.8413 and for 1.5, it's 0.9332.

The probability of filling a cup between 30 and 31 ounces is the difference between the cumulative probabilities of the two z-scores. So, the answer is 0.9332 - 0.8413 = 0.0919.

Therefore, the probability of the soft drink machine filling a cup between 30 to 31 ounces is 0.0919 or 9.19% when rounded to four decimal places.

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The probability of filling a cup between 30 and 31 ounces is approximately [tex]\(\boxed{0.0919}\)[/tex]

The probability of filling a cup between 30 and 31 ounces when the mean output is 28 ounces and the standard deviation is 2 ounces can be found using the Z-score formula for a normal distribution.

 First, we calculate the Z-score for 30 ounces:

[tex]\[ Z_{30} = \frac{X - \mu}{\sigma} = \frac{30 - 28}{2} = \frac{2}{2} = 1 \][/tex]

Next, we calculate the Z-score for 31 ounces:

[tex]\[ Z_{31} = \frac{X - \mu}{\sigma} = \frac{31 - 28}{2} = \frac{3}{2} = 1.5 \][/tex]

Now, we look up the probabilities corresponding to these Z-scores in the standard normal distribution table or use a calculator.

The probability of getting a value less than or equal to [tex]\( Z_{30} \)[/tex]is:

[tex]\[ P(Z \leq 1) \approx 0.8413 \][/tex]

 The probability of getting a value less than or equal to is:

[tex]\[ P(Z \leq 1.5) \approx 0.9332 \][/tex]

To find the probability of filling a cup between 30 and 31 ounces, we subtract the probability of filling up to 30 ounces from the probability of filling up to 31 ounces:

[tex]\[ P(30 < X < 31) = P(Z \leq 1.5) - P(Z \leq 1) \][/tex]

[tex]\[ P(30 < X < 31) \approx 0.0919 \][/tex]

 Rounded to four decimal places, the probability is 0.0919.

 Therefore, the probability of filling a cup between 30 and 31 ounces is approximately [tex]\(\boxed{0.0919}\)[/tex]

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

Answer: 0.5237

Step-by-step explanation:

Mean : [tex]\mu=192\text{ days}[/tex]

Standard deviation : [tex]\sigma = 12\text{ days}[/tex]

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

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

For x = 188 ​days ,

[tex]z=\dfrac{188-192}{12}\approx-0.33[/tex]

For x = 107 miles per day ,

[tex]z=\dfrac{107-92}{12}=1.25[/tex]

The P-value =[tex]P(-0.33<z<1.25)=P(z<1.25)-P(z<-0.33)[/tex]

[tex]0.8943502-0.3707=0.5236502\approx0.5237[/tex]

Hence, The probability that a randomly selected pregnancy lasts less than 188 days is approximately 0.5237.

Final answer:

To calculate the probability of a shopper spending less than 20 minutes in the store, the Z-score is found using the formula Z = (X - μ) / σ, resulting in a Z-score of -0.25, which corresponds to a probability of approximately 0.40.

Explanation:

The question asks for the probability that a randomly selected shopper will spend less than 20 minutes in a store, given that the average time spent is 22 minutes with a standard deviation of 8 minutes, and that these times are normally distributed. To find this probability, we use the Z-score formula:

Z = (X - μ) / σ

Where X is the value we are checking (20 minutes), μ is the mean (22 minutes), and σ is the standard deviation (8 minutes). Plugging in the numbers, we get:

Z = (20 - 22) / 8 = -0.25

Next, we look up the Z-score in a standard normal distribution table, or use a calculator with normal distribution functions, to find the probability that a Z-score is less than -0.25. This probability is approximately 0.40.

Answer:

D. 1.25 grams

Step-by-step explanation:

Half-life is: 0.004 sec

Time spent : 0.016 sec

Quantity = 20 gram

In order to find the material after 0.016 sec, we have to calculate how many number of half-lives have been passed

No. of half-lives passed = 0.016/0.004

= 4

The number of lives passed will be raised to the power of 0.5.

0.5 ^ 4 = 0.0625

The answer will be multiplied with the quantity we started with.

Remaining material is:

20*0.0625 = 1.25 grams

Hence, Option D is correct ..

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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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:

The percentage of the distribution is less than 75 is 97.72%.

Step-by-step explanation:

Given,

Mean of the distribution,

[tex]\mu=67[/tex]

Standard deviation,

[tex]\sigma = 4[/tex]

Thus, the z-score of the score 75,

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

[tex]=\frac{75-67}{4}[/tex]

[tex]=\frac{8}{4}[/tex]

[tex]=2[/tex]

With the help of z-score table,

[tex]P(x<75)=0.9772=97.72\%[/tex]

Hence, the percentage of the distribution is less than 75 is 97.72%.

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:

Final answer:

Stephanie cuts the rope into five 1-meter-long pieces by dividing the total length (5 meters) by the number of pieces (5).

Explanation:

If Stephanie cuts a five-meter-long piece of rope into five equal parts, we would divide the total length of the rope by the number of parts to find the length of each piece. In this case, the calculation would be:

So, the length of each part would be 5 meters ÷ 5, which equals 1 meter. Therefore, each piece of rope would be 1 meter long.

Answer: The cost of child's ticket = $10

The cost of adult ticket = $ 13

Step-by-step explanation:

Let x be the cost of a child ticket and y be the cost of an adult ticket.

Then According to the question, we have

[tex]6x+2y=86..........................(1)\\\\10x+3y=139.......................(2)[/tex]

Multiply equation (1) by 3 and equation (2) by 2, then we have

[tex]18x+6y=258.......................(1)\\\\20x+6y=278...........................(2)[/tex]

Subtract equation (1) from equation (2), we have

[tex]2x=20\\\\\Rightarrow\ x=10[/tex]

Substitute the value of x in equation (1), we get

[tex]60+2y=86\\\\\Rightarrow\ 2y=26\\\\\Rightarrow\ y=13[/tex]

Hence, the cost of child's ticket = $10

The cost of adult ticket = $ 13

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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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 with explanation:

The Inequality should be:

   [tex](1+x)^n\geq 1+n x[/tex]

Where, n and x are any integers.

⇒For, x= -1

L HS

 [tex]=[1+(-1)]^n\\\\=(0)^n\\\\=0[/tex]

R HS

→1+n × (-1)

=1-n

If, n is any Integer, then for, n=1

1-1=0

For, n=2

1-2= -1

....

So,   [tex](1+x)^n\geq 1+n x[/tex]

for, x=-1.

⇒For, x=1

L HS

 [tex]=[1+(1)]^n\\\\=(2)^n[/tex]

For, n=1

L H S=1

For, =2

L H S=4

R HS

→1+n × (1)

=1+n

If, n is any Integer, then for, n=1

1+1=2

For, n=2

1+2= 3

....

So,    [tex](1+x)^n\geq 1+n x[/tex]

for, x=1.

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]

Answer:

Step-by-step explanation:

You can't simplify the sum, but you can factor it.

t^2 + 2t - 3

(t  + 3)(t - 1 )

That's about all you can do.

Answer:

It is already in simplest form.

Step-by-step explanation:

It cannot be further simplified because it does not have any like terms.

Answer:

x = -5 sin (2t)

Step-by-step explanation:

k is the spring stiffness.  The unstretched length of the spring is L.

When the mass is added, the spring stretches to an equilibrium position of L+s, where mg = ks.  When the mass is displaced a distance x (where x is positive if the displacement is down and negative if it's up), the spring is stretched a total distance s + x.

There are two forces on the mass: weight and force from the spring.  Sum of the forces in the downward direction:

∑F = ma

mg − k(s + x) = ma

mg − ks − kx = ma

Since mg = ks:

-kx = ma

Acceleration is second derivative of position, so:

-kx = m d²x/dt²

Let's find k:

F = kx

400 = 2k

k = 200

We know that m = 50.  Substituting:

-200x = 50 d²x/dt²

-4x = d²x/dt²

d²x/dt² + 4x = 0

This is a linear second order differential equation of the form:

x" + ω² x = 0

The solution to this is:

x = A cos (ωt) + B sin (ωt)

Here, ω² = 4, so ω = 2.

x = A cos (2t) + B sin (2t)

We're given initial conditions that x(0) = 0 and x'(0) = -10 (remember that down is positive and up is negative).

Finding x'(t):

x' = -2A sin (2t) + 2B cos (2t)

Plugging in the initial conditions:

0 = A

-10 = 2B

Therefore:

x = -5 sin (2t)

Step-by-step explanation:

o - odd number

e - even number

n × e = e, n is either odd or even...rule 1

n - e = o, n must be odd...rule 2

n - e = e, n must be even...rule 3

n^2 = o, n must be odd...rule 4

n^2 = e, n must be even...rule 5

6n is even, no matter if n is odd or even following rule 1

if n^2 - 6n = o, n must be odd following rule 2

if n^2 = o, n must be odd following rule 4

Answer: a) 0.7814

b) False

Step-by-step explanation:

To find : The p-value of z , where

[tex]-1.23<z< +1.23[/tex]

[tex]P(-1.23<z< +1.23)=1-P(z<-1.23)\\\\=1-0.1093=0.7814[/tex]

Hence, the standard normal probability [tex]P(-1.23<z< +1.23)=0.7814[/tex]

In the formula used to convert "real-world" data values into z numbers, the standard deviation of the data is considered.

The formula to calculate the z score is [tex]z=\dfrac{x-\mu}{\sigma}[/tex].

If sample size (n) is given then , the [tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]

Hence, the statement is false.