(1 point) Find the general solution to the homogeneous differential equation. ????2y????????2−20????y????????+136y=0 Use c1 and c2 in your answer to denote arbitrary constants, and enter them as c1 and c2. y(????)= ?

Answer:

71.08% probability that pˆ takes a value between 0.17 and 0.23.

Step-by-step explanation:

We use the binomial approxiation to the normal to solve this question.

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

Can be approximated to a normal distribution, using the expected value and the standard deviation.

The expected value of the binomial distribution is:

[tex]E(X) = np[/tex]

The standard deviation of the binomial distribution is:

[tex]\sqrt{V(X)} = \sqrt{np(1-p)}[/tex]

Normal probability distribution

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

When we are approximating a binomial distribution to a normal one, we have that [tex]\mu = E(X)[/tex], [tex]\sigma = \sqrt{V(X)}[/tex].

In this problem, we have that:

[tex]p = 0.2, n = 200[/tex]. So

[tex]\mu = E(X) = np = 200*0.2 = 40[/tex]

[tex]\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{200*0.2*0.8} = 5.66[/tex]

In other words, find probability that pˆ takes a value between 0.17 and 0.23.

This probability is the pvalue of Z when X = 200*0.23 = 46 subtracted by the pvalue of Z when X = 200*0.17 = 34. So

X = 46

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{46 - 40}{5.66}[/tex]

[tex]Z = 1.06[/tex]

[tex]Z = 1.06[/tex] has a pvalue of 0.8554

X = 34

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{34 - 40}{5.66}[/tex]

[tex]Z = -1.06[/tex]

[tex]Z = -1.06[/tex] has a pvalue of 0.1446

0.8554 - 0.1446 = 0.7108

71.08% probability that pˆ takes a value between 0.17 and 0.23.

The probability that [tex]\( \hat{p} \)[/tex] takes a value between 0.17 and 0.23 is approximately 0.7108.

1. Given that 20% of adult women dye or highlight their hair, the true population proportion [tex]\( p \)[/tex] is 0.20.

2. We want to find the probability that a sample proportion [tex]\( \hat{p} \)[/tex] from a simple random sample (SRS) of size 200 falls within plus or minus 3 percentage points of this true value. In other words, we want to find [tex]\( P(0.17 < \hat{p} < 0.23) \)[/tex].

3. The standard error of [tex]\( \hat{p} \)[/tex] is given by:

[tex]\[ SE = \sqrt{\frac{p(1-p)}{n}} \][/tex]

where [tex]\( p = 0.20 \)[/tex] (the true population proportion) and [tex]\( n = 200 \)[/tex]  (the sample size).

4. Calculate the standard error:

[tex]\[ SE = \sqrt{\frac{0.20 \times (1-0.20)}{200}} = \sqrt{\frac{0.20 \times 0.80}{200}} \][/tex]

[tex]\[ SE = \sqrt{\frac{0.16}{200}} = \sqrt{0.0008} \approx 0.0283 \][/tex]

5. Next, we find the z-scores corresponding to the values 0.17 and 0.23 using the standard normal distribution table:

[tex]\[ z_{0.17} = \frac{0.17 - 0.20}{0.0283} \approx -1.0601 \][/tex]

[tex]\[ z_{0.23} = \frac{0.23 - 0.20}{0.0283} \approx 1.0601 \][/tex]

6. Using the z-scores, we find the corresponding probabilities from the standard normal distribution table:

[tex]\[ P(\hat{p} < 0.17) \approx P(Z < -1.0601) \approx 0.1446 \][/tex]

[tex]\[ P(\hat{p} < 0.23) \approx P(Z < 1.0601) \approx 0.8554 \][/tex]

7. Therefore, the probability that [tex]\( \hat{p} \)[/tex] takes a value between 0.17 and 0.23 is approximately:

[tex]\[ P(0.17 < \hat{p} < 0.23) = P(\hat{p} < 0.23) - P(\hat{p} < 0.17) \][/tex]

[tex]\[ \approx 0.8554 - 0.1446 = 0.7108 \][/tex]

8. Alternatively, we can find this probability directly using the cumulative distribution function (CDF) of the standard normal distribution:

[tex]\[ P(0.17 < \hat{p} < 0.23) = P(-1.0601 < Z < 1.0601) \][/tex]

[tex]\[ \approx \Phi(1.0601) - \Phi(-1.0601) \][/tex]

[tex]\[ \approx 0.8554 - 0.1446 = 0.7108 \][/tex]

9. Therefore, the probability that[tex]\( \hat{p} \)[/tex] takes a value between 0.17 and 0.23 is approximately 0.7108.

Answer:

a) (g(x), f(u)) = ( 7*√x , e^u )

b)   y ' = 3.5 * e^(7*√x) / √x

Step-by-step explanation:

Given:

- The given function:

                                       y = e^(7*√x)

Find:

- Express the given function as a composite of f(g(x)). Where, u = g(x) and y = f(u).

- Express the derivative of y, y'?

Solution:

- We will assume the exponent of  the natural log to be the u. So u is:

                                     u = g(x) = 7*√x

- Then y is a function of u as follows:

                                     y = f(u) = e^u

- The composite function is as follows:

                                    (g(x), f(u)) = ( 7*√x , e^u )

- The derivative of y is such that:

                                    y = f(g(x))

                                    y' = f' (g(x) ) * g'(x)

                                    y' = f'(u) * g'(x)

                                    y' = e^u* 3.5 / √x

- Hence,

                                   y ' = 3.5 * e^(7*√x) / √x

Answer:

b. Poisson probability distribution

Step-by-step explanation:

The Poisson distribution is the discrete probability distribution of the number of events occurring in a given time period. The poisson distribution is also use when few large demand is expected.

In this question; poisson distribution is use to find the probability of 10 customers arrivals in an hour at a service station.

To find the probability of 10 customer arrivals in an hour at a service station, one would use the Poisson probability distribution, which calculates the probability of a certain number of events happening in a set period of time.

The correct answer is b. Poisson probability distribution. This type of distribution is often used to calculate the probability of a certain number of events happening in a set period of time. In this case, the 'events' are the arrivals of customers at a service center within an hour.

Here's a very simplified version of the steps to calculate a Poisson probability:
Step 1: Identify the average rate (λ) - this is the average number of times the event is happening per unit of measure (in your case, customer arrivals per hour).
Step 2: Use the formula for Poisson probability, which is P(x; λ) = e^-λ * λ^x / x! Where 'x' is the actual number of successes that result from the experiment, 'e' is approx 2.71828 and '!' denotes a factorial.

So, if we knew the average rate of customer arrivals, we could easily apply it to this formula to get the probability of 10 customer arrivals in an hour.

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Answer:

dx/dt  =  0,04 m/sec

Step-by-step explanation:

Area of the circle is:

A(c) =π*x²      where   x is a radius of the circle

Applying differentiation in relation to time we get:

dA(c)/dt   =  π*2*x* dx/dt    

In this equation we know:

dA(c)/dt  = 0,5 m²/sec

And are looking for dx/dt then

0,5  = 2*π*x*dx/dt    when the area of the sheet is 12 m²  (1)

When  A(c) = 12 m²      x = ??

A(c)  =  12  =  π*x²      ⇒    12  =  3.14* x²    ⇒  12/3.14  =  x²

x²  = 3,82     ⇒   x  = √3,82    ⇒  x = 1,954 m

Finally plugging ths value in equation (1)

0,5  = 6,28*1,954*dx/dt

dx/dt  =  0,5 /12.28

dx/dt  =  0,04 m/sec

The rate at which the radius is decreasing when the area of the sheet is 12 m² is; dr/dt = 0.041 m/s

We are given;

Area of sheet; A = 12 m²

Rate of change of area; dA/dt = 0.5 m²/s

Now, formula for area of the circular sheet is given as;

A = πr²

Thus; 12 = πr²

r = √(12/π)

r = 1.9554 m

Now, we want to find the rate at which the radius is decreasing and so we differentiate both sides of the area formula with respect to t;

dA/dt = 2πr(dr/dt)

Thus;

0.5 = 2π × 1.9554(dr/dt)

dr/dt = 0.5/(2π × 1.9554)

dr/dt = 0.041 m/s

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Answer:

a) 0.85

b) 0.15

c) 0.40

d) 0.70

Step-by-step explanation:

P(A) = 0.55

P(B) = 0.45

P(A n B) = 0.15

a) Probability of A or B occurring = P(A u B) = P(A) + P(B) - P(A n B) = 0.55 + 0.45 - 0.15 = 0.85

b) Probability of A and B occurring = P(A n B) = 0.15

c) Probability of just A occurring = P(A n B') = P(A) - P(A n B) = 0.55 - 0.15 = 0.40

d) Probability of just A or just B occurring = P(A n B') + P(A' n B) = 0.4 + (0.45 - 0.15) = 0.4 + 0.3 = 0.70

Answer:

The minimum score required for an A grade is 86.

Step-by-step explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

[tex]\mu = 72.1, \sigma = 9.5[/tex]

Find the minimum score required for an A grade.

Top 7%, which is the value of X when Z has a pvalue of 1-0.07 = 0.93. So it is X when Z = 1.475. So

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]1.475 = \frac{X - 72.1}{9.5}[/tex]

[tex]X - 72.1 = 1.475*9.5[/tex]

[tex]X = 86[/tex]

The minimum score required for an A grade is 86.

Answer:

[tex]p \sim N(p,\sqrt{\frac{p(1-p)}{n}})[/tex]

For this case we know this:

[tex] n=37 ,  p=0.2[/tex]

We can find the standard error like this:

[tex] SE = \sqrt{\frac{\hat p (1-\hat p)}{n}}= \sqrt{\frac{0.2*0.8}{37}}= 0.0658[/tex]

So then our random variable can be described as:

[tex] p \sim N(0.2, 0.0658)[/tex]

Let's suppose that the question on this case is find the probability that the population proportion would be higher than 0.4:

[tex] P(p>0.4)[/tex]

We can use the z score given by:

[tex] z = \frac{p -\mu_p}{SE_p}[/tex]

And using this we got this:

[tex] P(p>0.4) = 1-P(z< \frac{0.4-0.2}{0.0658}) = 1-P(z<3.04) = 0.0012[/tex]

And we can find this probability using the Ti 84 on this way:

2nd> VARS> DISTR > normalcdf

And the code that we need to use for this case would be:

1-normalcdf(-1000, 3.04; 0;1)

Or equivalently we can use:

1-normalcdf(-1000, 0.4; 0.2;0.0658)

Step-by-step explanation:

We need to check if we can use the normal approximation:

[tex] np = 37 *0.2 = 7.4 \geq 5[/tex]

[tex] n(1-p) = 37*0.8 = 29.6\geq 5[/tex]

We assume independence on each event and a random sampling method so we can conclude that we can use the normal approximation and then ,the population proportion have the following distribution :

[tex]p \sim N(p,\sqrt{\frac{p(1-p)}{n}})[/tex]

For this case we know this:

[tex] n=37 ,  p=0.2[/tex]

We can find the standard error like this:

[tex] SE = \sqrt{\frac{\hat p (1-\hat p)}{n}}= \sqrt{\frac{0.2*0.8}{37}}= 0.0658[/tex]

So then our random variable can be described as:

[tex] p \sim N(0.2, 0.0658)[/tex]

Let's suppose that the question on this case is find the probability that the population proportion would be higher than 0.4:

[tex] P(p>0.4)[/tex]

We can use the z score given by:

[tex] z = \frac{p -\mu_p}{SE_p}[/tex]

And using this we got this:

[tex] P(p>0.4) = 1-P(z< \frac{0.4-0.2}{0.0658}) = 1-P(z<3.04) = 0.0012[/tex]

And we can find this probability using the Ti 84 on this way:

2nd> VARS> DISTR > normalcdf

And the code that we need to use for this case would be:

1-normalcdf(-1000, 3.04; 0;1)

Or equivalently we can use:

1-normalcdf(-1000, 0.4; 0.2;0.0658)

Answer:

$4848.8

Step-by-step explanation:

(1 + 0.05)⁵ = 1.276

FV = 3800 × 1.276

= 4848.8

Final answer:

The future value of $3,800 invested for five years at a 5% interest rate, using the formula [tex]FV = PV[/tex] x [tex](1 + r)^t[/tex], is approximately $4,849.07.

Explanation:

You are asking how to calculate the future value of an amount of money when invested at a certain interest rate over a set period of time. Specifically, you want to know the future value of $3,800 invested for five years at a 5% interest rate.

To calculate the future value (FV) of money we use the formula:
[tex]FV = PV X (1 + r)^t[/tex]

Where:

PV is the present value or initial amount ($3,800)

r is the annual interest rate (5%, or as a decimal, 0.05)

t is the time in years the money is invested (5 years)

Using the formula, we get:
FV = $3,800 x (1 + 0.05)5

FV = $3,800 x (1.276281)

FV = $3,800 x 1.276 (rounded to three decimal places)

FV = $4,849.07 (rounded to two decimal places)

Hence, the future value of $3,800 invested for five years at a 5% interest rate would be approximately $4,849.07.

Answer:4y^2-10y-1

Step-by-step explanation:

Answer: The second option, x^4 - 8x^2 - 16.

Step-by-step explanation:

Polynomials in standard form start with the highest degree, from greatest to least exponent. After all terms with exponents are in order, alphabetical variables are next. In this case there's only x. Last are constant terms, which are by itself, with no variable next to it/an exponent to the right of it.

x^4 - 8x^2 - 16 is in standard form because it follows the criteria above. 4 is the highest degree since it's the highest exponent in the polynomial expression, which is why it starts off with x^4. Other terms with lesser exponents are next. In this case, it's 8x^2 with the less exponent of 2. Finally, it ends with your constant term, -16.

The standard form of the polynomial is,

⇒ x⁴ - 8x² - 16

Given that,

All the polynomials are,

⇒ 5 - 2x

⇒ x⁴ - 8x² - 16

⇒ 5x³ + 4x⁴ - 3x + 1

Since we know that,

In standard form, a polynomial is arranged in descending order of the exponents of its terms.

This means that the term with the highest degree is listed first, followed by the terms with lower degrees.

Hence, Based on this definition, the polynomial in standard form among the options is,

⇒ x⁴ - 8x² - 16

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Answer:

0.0082 = 0.82% probability that he will pass

Step-by-step explanation:

For each question, there are only two possible outcomes. Either the students guesses the correct answer, or he guesses the wrong answer. The probability of guessing the correct answer for a question is independent of other questions. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.

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

And p is the probability of X happening.

In this problem we have that:

[tex]n = 14, p = 0.3[/tex].

If the student makes knowledgeable guesses, what is the probability that he will pass?

He needs to guess at least 9 answers correctly. So

[tex]P(X \geq 9) = P(X = 9) + P(X = 10) + P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14)[/tex]

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]P(X = 9) = C_{14,9}.(0.3)^{9}.(0.7)^{5} = 0.0066[/tex]

[tex]P(X = 10) = C_{14,10}.(0.3)^{10}.(0.7)^{4} = 0.0014[/tex]

[tex]P(X = 11) = C_{14,11}.(0.3)^{11}.(0.7)^{3} = 0.0002[/tex]

[tex]P(X = 12) = C_{14,12}.(0.3)^{12}.(0.7)^{2} = 0.000024[/tex]

[tex]P(X = 13) = C_{14,13}.(0.3)^{13}.(0.7)^{1} = 0.000002[/tex]

[tex]P(X = 14) = C_{14,14}.(0.3)^{14}.(0.7)^{0} \cong 0 [/tex]

[tex]P(X \geq 9) = P(X = 9) + P(X = 10) + P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14) = 0.0066 + 0.0014 + 0.0002 + 0.000024 + 0.000002 = 0.0082[/tex]

0.0082 = 0.82% probability that he will pass

Jack inherited a perpetuity-due, which he exchanged for a 25-year annuity-due. The present value of both are equal and based on different interest rates over the 25 years. The annual payment X of the annuity can be calculated using the formula for the present value of an annuity and the present value of the inherited perpetuity-due.

This problem is about financial mathematics, specifically involving perpetuities and annuities. In a perpetuity-due, the payments are made at the beginning of each period indefinitely. An annuity-due is similar, but the payments only last a specified number of years.

The present value (PV) of a perpetuity-due with annual payments P and annual interest rate r is calculated by PV = P / r. Given that P is $15,000 and r is 10%, the present value of the perpetuity-due that Jack inherits is $150,000.

When Jack swaps this for a 25-year annuity-due, with the first 10 years at 10% interest and the next 15 years at 8% interest, we calculate the annual payment X using the formula for the present value of an annuity. This formula involves dividing the total present value by the sum of the present value factors for each year at the respective interest rates.

Consequently, X can be calculated as follows: X = (PV of perpetuity-due) / ∑ (discount factors for each year). A detailed calculation will give the exact value of X.

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Answer:

Figure out the various probabilities first, that will make the rest of the questions easier:

P(discovered) = .7

P(not discovered) = 1 - .7 = .3

P(locator|discovered) = .6

P(no locator|discovered) = 1 - .6 = .4

P(locator|not discovered) = 1 - .9 = .1

P(no locator|not discovered) = .9

P(discovered and locator) = .7 * .6 = .42

P(discovered and no locator) = .7 * .4 = .28

P(not discovered and locator) = .3 * .1 = .03

P(not discovered and no locator) = .3 * .9 = .27

a) The total probability that an aircraft has a locator is .42 + .03 = .45. So the probability it will not be discovered, given it has a locator, is .03/.45 = .067

b) The total probability that an aircraft does not have a locator is .28 + .27 = .55. So the probability it will be discovered, given it does not have a locator, is .28/.55 = .509

c) Probability that 7 are discovered = C(10,7) * P(discovered|locator)^7 * P(not discovered|locator)^3

We already figured out P(not discovered|locator) = .067, so P(discovered|locator) = 1-.067 = .933. C(10,7) = 10*9*8, so we can compute total probability: 10*9*8 * .933^7 * .067^3 = .133

Step-by-step explanation:

Answer: Length of arc DB is 14π feet.

Step-by-step explanation:

The sum of the angles on a straight line is 180 degrees. This means that

m∠DAB + m∠CAB = 180

m∠DAB + 40 = 180

m∠DAB = 180 - 40

m∠DAB = 140°

The formula for determining the length of an arc is expressed as

Length of arc = θ/360 × 2πr

Where

θ represents the central angle.

r represents the radius of the circle.

π is a constant whose value is 3.14

From the information given,

r = 18 feet

θ = 140°

Therefore,

Length of arc DB = 140/360 × 2 × π × 18

Length of arc DB = 14π feet

Answer:

Given p = 0.07 as the probability that someone is a universal donor

In case of Geometric Distribution, Probability of  getting the first success on nth trial is given by

[tex]P (X=n) = p (1-p) ^ {n-1}[/tex]

where p is the probability of success on any one trial and (1-p) shows the probability of failure.

So the probability of the first subject to be a universal blood donor will be the seventh person is

[tex]P (X=7) = 0.07 (1-0.07) ^ {7-1} = 0.07 (0.93) ^ 6 = 0.07*0.647 = 0.0453[/tex]

So the final probability is 0.0453

Answer:

a) Probability that a randomly selected commuter will spend more than 7 minutes waiting for GO-D.O.T = P(7 < x ≤ 20) = 0.65

b) Standard deviation of the uniform distribution = 5.77 minutes

c) Probability that a randomly selected commuter will spend longer than 10 minutes but no more than 17 minutes waiting for the GO-D.O.T = P(10 < x < 17) = 0.35

d) average waiting time for the uniform distribution = 10 minutes.

Step-by-step explanation:

This is a uniform distribution problem with lower limit of 0 minute and upper limit of 20 minutes.

a = 0, b = 20

Probability = f(x) = [1/(b-a)] ∫ dx (with the definite integral evaluated between the two intervals whose probability is required.

a) Probability that a randomly selected commuter will spend more than 7 minutes waiting for GO-D.O.T

P(7 < x ≤ 20) = f(x) = [1/(b-a)] ∫²⁰₇ dx

P(7 < x ≤ 20) = (20-7)/(20-0) = (13/20) = 0.65

b) Standard deviation of the uniform distribution

Standard deviation of a uniform distribution is given as

σ = √[(b-a)²/12]

σ = √[(20-0)²/12]

σ = √[20²/12]

σ = 5.77 minutes

c) Probability that a randomly selected commuter will spend longer than 10 minutes but no more than 17 minutes waiting for the GO-D.O.T = P(10 < x < 17)

P(10 < x < 17) = (17-10)/(20-0)

P(10 < x < 17) = (7/20) = 0.35

d) The average waiting time.

The average of a uniform distribution = (b+a)/2

Average waiting time = (20+0)/2

Average waiting time = 10 minutes

Hope this Helps!!!

Answer:

a) 50%

b) 68%

c) 99%

Step-by-step explanation:

for a standard normal curve ,

a) since the standard normal curve is symmetric and centred around μ , 50% of the curve lies at the left of μ and 50% lies to the right

b) according to the 68-95-99 rule,  68% of the standard normal curve lies from μ − σ and μ + σ

c) from the same rule , 99% of the standard normal curve lies from μ − 3σ and μ + 3σ

Answer:

a) 50%

b) 68%

c) 99.7%

Step-by-step explanation:

The Empirical Rule states that, for a normally distributed random variable:

68% of the measures are within 1 standard deviation of the mean.

95% of the measures are within 2 standard deviation of the mean.

99.7% of the measures are within 3 standard deviations of the mean.

The normal distribution is also symmetric, which means that 50% of the measures are below the mean and 50% are above.

In this problem, we have that:

Mean μ

Standard deviation σ

Area under the normal curve = percentage

a) What percentage of the area under the normal curve lies to the left of μ?

Normal distribution is symmetric, so the answer is 50%.

(b) What percentage of the area under the normal curve lies between μ − σ and μ + σ?  

Within 1 standard deviation of the mean, so 68%.

(c) What percentage of the area under the normal curve lies between μ − 3σ and μ + 3σ?

Within 3 standard deviation of the mean, so 99.7%.

Answer:

Choices={SB,SR,MB,MR,LS,LM,XLB,XLR}

8 combined choices

Step-by-step explanation:

Combinations

We'll define two sets of options for the windbreaker jackets, one for the sizes and another for the colors. Being S=small, M=medium, L=large, and XL=extra large, then

Z={S,M,L,XL}

is the set of possible sizes for the windbreaker jackets. Now, being B=blue and R=red, the set of colors is

C={B,R}

The combined choices are found by the cartesian product of ZxC:

Choices={SB,SR,MB,MR,LS,LM,XLB,XLR}

Where MB, for example, is Medium-Blue

That is the sample space for all the possible combinations, there are 8 in total

According to the Counting Principle in Mathematics, there are 8 different combinations of sizes and colors for the windbreaker jackets available at the retail store.

The retail store offers windbreaker jackets in four different sizes: small, medium, large, and extra large. Each of these sizes is available in two colors: blue and red. Therefore, using a concept in mathematics known as the Counting Principle, we can ascertain the number of combinations. The Counting Principle states that if one event can occur in m ways and another can occur in n ways, then the number of ways that both events can occur is m*n.

So in this scenario, we have 4 sizes (small, medium, large, extra-large) and 2 colors (blue, red). Applying the Counting Principle, there are a total of 4 * 2 = 8 different combinations of jackets that can be purchased from the store.

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Answer:

A≈309.02cm² Step by step

Answer:

a) [tex]P(X<80)=P(\frac{X-\mu}{\sigma}<\frac{80-\mu}{\sigma})=P(Z<\frac{80-72.6}{4.78})=P(z<1.548)[/tex]

And we can find this probability using the normal standard distirbution or excel and we got:

[tex]P(z<1.548)=0.939[/tex]

And that correspond to 93.9 %

b) [tex]P(60<X<80)=P(\frac{60-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{80-\mu}{\sigma})=P(\frac{60-72.6}{4.78}<Z<\frac{80-72.6}{4.78})=P(-2.636<z<1.548)[/tex]

And we can find this probability with this difference:

[tex]P(-2.636<z<1.548)=P(z<1.548)-P(z<-2.636)[/tex]

And in order to find these probabilities we can find tables for the normal standard distribution, excel or a calculator.  

[tex]P(-2.636<z<1.548)=P(z<1.548)-P(z<-2.636)=0.939-0.0042=0.935 [/tex]

So we have approximately 93.5%

c) [tex]z=1.64<\frac{a-72.6}{4.78}[/tex]

And if we solve for a we got

[tex]a=72.6 +1.64*4.78=80.439[/tex]

So the value of velocity that separates the bottom 95% of data from the top 5% is 80.439.  

d) [tex]P(X>70)=P(\frac{X-\mu}{\sigma}>\frac{70-\mu}{\sigma})=P(Z>\frac{70-72.6}{4.78})=P(z>-0.544)[/tex]

And we can find this probability using the complement rule, normal standard distirbution or excel and we got:

[tex]P(z>-0.544)=1-P(z<-0.544) = 1-0.293=0.707 [/tex]

And that correspond to 70.7 %

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Part a

Let X the random variable that represent the vehicles speeds of a population, and for this case we know the distribution for X is given by:

[tex]X \sim N(72.6,4.78)[/tex]  

Where [tex]\mu=72.6[/tex] and [tex]\sigma=4.78[/tex]

We are interested on this probability

[tex]P(X<80)[/tex]

And the best way to solve this problem is using the normal standard distribution and the z score given by:

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

If we apply this formula to our probability we got this:

[tex]P(X<80)=P(\frac{X-\mu}{\sigma}<\frac{80-\mu}{\sigma})=P(Z<\frac{80-72.6}{4.78})=P(z<1.548)[/tex]

And we can find this probability using the normal standard distirbution or excel and we got:

[tex]P(z<1.548)=0.939[/tex]

And that correspond to 93.9 %

Part b

We want this probability

[tex]P(60<X<80)=P(\frac{60-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{80-\mu}{\sigma})=P(\frac{60-72.6}{4.78}<Z<\frac{80-72.6}{4.78})=P(-2.636<z<1.548)[/tex]

And we can find this probability with this difference:

[tex]P(-2.636<z<1.548)=P(z<1.548)-P(z<-2.636)[/tex]

And in order to find these probabilities we can find tables for the normal standard distribution, excel or a calculator.  

[tex]P(-2.636<z<1.548)=P(z<1.548)-P(z<-2.636)=0.939-0.0042=0.935 [/tex]

So we have approximately 93.5%

Part c

For this part we want to find a value a, such that we satisfy this condition:

[tex]P(X>a)=0.05[/tex]   (a)

[tex]P(X<a)=0.95[/tex]   (b)

Both conditions are equivalent on this case. We can use the z score again in order to find the value a.  

As we can see on the figure attached the z value that satisfy the condition with 0.95 of the area on the left and 0.05 of the area on the right it's z=1.64. On this case P(Z<1.64)=0.95 and P(z>1.64)=0.05

If we use condition (b) from previous we have this:

[tex]P(X<a)=P(\frac{X-\mu}{\sigma}<\frac{a-\mu}{\sigma})=0.95[/tex]  

[tex]P(z<\frac{a-\mu}{\sigma})=0.95[/tex]

But we know which value of z satisfy the previous equation so then we can do this:

[tex]z=1.64<\frac{a-72.6}{4.78}[/tex]

And if we solve for a we got

[tex]a=72.6 +1.64*4.78=80.439[/tex]

So the value of velocity that separates the bottom 95% of data from the top 5% is 80.439.  

Part d

[tex]P(X>70)[/tex]

And the best way to solve this problem is using the normal standard distribution and the z score given by:

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

If we apply this formula to our probability we got this:

[tex]P(X>70)=P(\frac{X-\mu}{\sigma}>\frac{70-\mu}{\sigma})=P(Z>\frac{70-72.6}{4.78})=P(z>-0.544)[/tex]

And we can find this probability using the complement rule, normal standard distirbution or excel and we got:

[tex]P(z>-0.544)=1-P(z<-0.544) = 1-0.293=0.707 [/tex]

And that correspond to 70.7 %

Answer:

[tex]\theta= \frac{\pi}{2} +\pi \cdot i[/tex], for all [tex]i = \mathbb{Z} \cup\{0\}[/tex]

Step-by-step explanation:

The velocity vector is found by deriving the position vector depending on the time:

[tex]\dot r(t)= v (t) = 3 \cdot i +\sqrt{2} \cdot j + 2\cdot t \cdot k[/tex]

In turn, acceleration vector is found by deriving the velocity vector depending on time:

[tex]\ddot r(t) = \dot v(t) = a(t) = 2 \cdot k[/tex]

Velocity and acceleration vectors at [tex]t = 0[/tex] are:

[tex]v(0) = 3\cdot i + \sqrt{2} \cdot j\\a(0) = 2 \cdot k\\[/tex]

Norms of both vectors are, respectively:

[tex]||v(0)||\approx 3.317\\||a(0)|| \approx 2[/tex]

The angle between both vectors is determined by using the following characteristic of a Dot Product:

[tex]\theta = \cos^{-1}(\frac{v(0) \bullet a(0)}{||v(0)||\cdot ||a(0)||})[/tex]

Given that cosine has a periodicity of [tex]\pi[/tex]. There is a family of solutions with the form:

[tex]\theta= \frac{\pi}{2} +\pi \cdot i[/tex], for all [tex]i = \mathbb{Z} \cup\{0\}[/tex]

Final answer:

π/2 radians, indicating the vectors are perpendicular at that instant.

Explanation:

The angle between the velocity and acceleration vectors at a given time can be found by first determining the velocity (νt) and acceleration (ν2t) vectors as the first and second derivatives of the position vector r(t). At time t=0, these derivatives can be calculated and then used to find the angle through the dot product and magnitude of these vectors.

For the given position vector

r(t) = (3t+9)i + (√2t)j + (t2)k,

the velocity vector v(t) is obtained by differentiating each component of r(t) with respect to time t, which gives

v(t) = (3)i + (√2)j + (2t)k.

Similarly, acceleration a(t) is the derivative of velocity v(t), which results in a(t) = (0)i + (0)j + (2)k.

At t=0, v(0) = (3)i + (√2)j and a(0) = (2)k. The angle θ between v(0) and a(0) is given by the cosine of the angle between the two vectors, which is calculated using the dot product formula:

θ = cos-1((v ⋅ a) / (|v||a|)).

Here, (v ⋅ a) is the dot product of v(0) and a(0), and |v| and |a| are the magnitudes of v(0) and a(0), respectively.

Since v(0) and a(0) are perpendicular at t=0, their dot product is 0, and the magnitudes of v(0) and a(0) do not affect the angle. Therefore, the angle θ is simply cos-1(0), which is π/2 radians, indicating the vectors are perpendicular.

a) The marginal cost function is [tex]\[C'(x) = 12 - 0.2x + 0.0015x^2\][/tex].

b) The value C'(500) = 287 predicts that for every additional yard of fabric produced at this level (500 yards), the cost will increase by $287.

c) The cost of manufacturing the 501st yard of fabric is approximately $45,000, which is approximately equal to the marginal cost C'(500)

a)

To find the marginal cost function, we need to take the derivative of the cost function C(x) with respect to x:

[tex]\[C(x) = 1500 + 12x - 0.1x^2 + 0.0005x^3\][/tex]

Taking the derivative:

[tex]\[C'(x) = \frac{d}{dx}(1500 + 12x - 0.1x^2 + 0.0005x^3)\][/tex]

[tex]\[C'(x) = 12 - 0.2x + 0.0015x^2\][/tex]

b)

To find [tex]\(C'(500)\)[/tex] substitute x = 500 into the marginal cost function:

[tex]\[C'(500) = 12 - 0.2(500) + 0.0015(500^2)\][/tex]

[tex]\[C'(500) = 12 - 100 + 375\][/tex]

[tex]\[C'(500) = 287\][/tex]

The value C'(500) = 287 represents the rate at which costs are increasing with respect to the production level when x = 500. It predicts that for every additional yard of fabric produced at this level (500 yards), the cost will increase by $287.

c)

To compare C'(500) with the cost of manufacturing the 501st yard of fabric, calculate C(501) - C(500):

[tex]\[C(501) - C(500) = (1500 + 12(501) - 0.1(501)^2 + 0.0005(501)^3) - (1500 + 12(500) - 0.1(500)^2 + 0.0005(500)^3)\][/tex]

Calculate the difference:

[tex]\[C(501) - C(500) = -45,000\][/tex]

Therefore, the cost of manufacturing the 501st yard of fabric is approximately $45,000, which is approximately equal to the marginal cost C'(500) when rounded to four decimal places.

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(a) Marginal cost function: C'(x) = 12 - 0.2x + 0.0015x².

(b) C'(500) predicts a $287 cost increase for the 500th yard.

(c) Cost for the 501st yard is approximately $512, slightly higher than C'(500).

(a) To find the marginal cost function, we need to calculate the derivative of the cost function C(x) with respect to x:

C(x) = 1500 + 12x - 0.1x² + 0.0005x³

C'(x) = d/dx [1500 + 12x - 0.1x² + 0.0005x³]

C'(x) = 12 - 0.2x + 0.0015x²

So, the marginal cost function is C'(x) = 12 - 0.2x + 0.0015x².

(b) To find C'(500), we plug in x = 500 into the marginal cost function:

C'(500) = 12 - 0.2(500) + 0.0015(500)²

C'(500) = 12 - 100 + 375

C'(500) = 287

C'(500) represents the rate at which costs are increasing with respect to the production level when 500 yards of fabric are produced. In this case, it predicts that the cost is increasing at a rate of $287 per yard for the 500th yard produced.

(c) To compare C'(500) with the cost of manufacturing the 501st yard of fabric, we need to calculate C(501) - C(500):

C(501) - C(500) = [1500 + 12(501) - 0.1(501)² + 0.0005(501)³] - [1500 + 12(500) - 0.1(500)² + 0.0005(500)³]

C(501) - C(500) = [1500 + 6012 - 25005 + 627507] - [1500 + 6000 - 25000 + 625000]

C(501) - C(500) = [627012] - [626500]

C(501) - C(500) = 512

So, the cost of manufacturing the 501st yard of fabric is $512, which is approximately equal to C'(500), which was calculated as $287. This means that the cost increased by approximately $287 for the 500th yard and by $512 for the 501st yard.

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Answer:

2 contracts

Step-by-step explanation:

Break even point refers to the number of units or sales that needs to be generated for the company to make neither a profit nor a loss.

This means that at the break even point, sales is equivalent to the cost incurred (both fixed and variable).

Let the number of contracts that must be signed to break even be s

The rent is a fixed cost while the cost associated with each contract is variable.

5687.1s = 7000 + 1260.7s

5687.1s - 1260.7s = 7000

4426.4 s = 7000

s = 1.58

≈ 2

She must facilitate 2 contracts each month to break even.

Answer:

(-1,9)

Step-by-step explanation:

Answer:

Based on the graph below, what is the total number of solutions to the equation f(x)= g(x) will be 3.

Step-by-step explanation:

The intersection points of both graphs would be the total number of solutions to the equation f(x)= g(x).

From the given diagram, it is clear that both the graphs intersect at three locations points or intersection points. The approximations locations of The intersection points of  both graphs are

Therefore, based on the graph below, what is the total number of solutions to the equation f(x)= g(x) will be 3.

Answer:A researcher uses probability to decide whether the sample she obtained is likely to be a sample from a particular population.

Step-by-step explanation: Inferential statistics is a Statistical process used to compare two or more samples or treatments.

Probability helps in inferential statistics to decide whether the sample obtained is likely from the population of interest.

Inferential statistics use data obtained from the sample of interest in a research to compare the treatment or samples.Through Inferential statistics researchers make conclusions about the entire population.

Probability in inferential statistics is used to make inferences about a population based on sample data. It provides the foundation for statistical methods such as confidence intervals and hypothesis testing to evaluate the accuracy of the sample in representing the population.

Probability in inferential statistics is critical in helping researchers make inferences about a population from a sample. When researchers collect data from a sample, they use probability theory to deduce how likely it is that their observations are reflective of the entire population or occured by chance. Inferential statistical methods, such as confidence intervals and hypothesis testing, leverage probability to make these determinations.

Probability enables statisticians to evaluate the accuracy of the sample data in representing the population, decide how confident they can be about their inferences, and test the validity of existing hypotheses about the population parameters based on sample data.

For instance, if an inferential statistical test indicates that the likelihood of obtaining the observed sample results by chance is only 5%, researchers can infer there is a 95% probability that the sample accurately reflects the population, supporting the hypothesis being tested. Therefore, probability is used to determine how much confidence researchers can have in their sample data when making generalizations about a larger group.

Answer:

note:

solution is attached in word form due to error in mathematical equation. furthermore i also attach Screenshot of solution in word due to different version of MS Office please find the attachment

Answer: probability that a randomly selected newborn is low-weight is 0.1038

Step-by-step explanation:

Since Birth weights of full-term babies in a certain area are normally distributed m, we would apply the formula for normal distribution which is expressed as

z = (x - µ)/σ

Where

x = birth weights of full-term babies.

µ = mean weight

σ = standard deviation

From the information given,

µ = 7.13 pounds

σ = 1.29 pounds

The probability that a randomly selected newborn is low-weight is expressed as

P(x ≤ 5.5)

For x = 5.5

z = (5.5 - 7.13)/1.29 = - 1.26

Looking at the normal distribution table, the probability corresponding to the z score is 0.1038

P(x ≤ 5.5) = 0.1038

Answer:

Each player gets 6 tennis balls.

4 × t = 24

Step-by-step explanation:

The total number of tennis balls in the basket is, 24 balls.

The number of tennis players is, 4 players.

On equally dividing all the balls among the 4 players, each player gets,

[tex]No.\ of\ balls\ each\ player\ gets=\frac{No.\ of\ balls}{No.\ of\ players} \\=\frac{24}{4} \\=6[/tex]

Thus, each player gets 6 tennis balls.

If we want to represent the equation of the number of players and number of balls, the equation will be,

4 × t = 24

t = number of tennis balls each player gets.

Final answer:

To divide 24 tennis balls evenly among four players, we use the equation 4t = 24, where 't' is the number of balls per player. By dividing 24 by 4, we find that each player receives 6 tennis balls.

Explanation:

The student is asking how to divide 24 tennis balls evenly among four players, which is a basic division problem in mathematics. The correct mathematical equation to represent this situation is C. 4t = 24, where 't' represents the number of tennis balls each player gets. To find the value of 't', we divide the total number of balls, 24, by the number of players, 4.

We perform the division as follows:

Write down the equation: 4t = 24.

Divide both sides of the equation by 4 to solve for 't': t = 24 / 4.

Calculate the result: t = 6.

Each player gets 6 tennis balls.

Answer:see the pictures attached

Step-by-step explanation: