Solve the following problems (show your work). (a) (1 point). Give example of 2 x 2 non-zero matrices A, B, C such that AB = AC but B+C. (b) (1 point). Write 2 x 2 invertible matrices A and B such that A+B is not invertible. (c) (2 points). Write 3 x 3 singular matrices A and B such that A - B is non-singular.

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:

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:4y^2-10y-1

Step-by-step explanation:

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:

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) We have the following distribution [tex] X \sim N(\mu =650, \sigma =10)[/tex]

And we want to calculate:

[tex] P(X>654) [/tex]

And in order to calclate this in the ti 84 we can use the following code

2nd> Vars>DISTR

And then we need to use the following code:

1-normalcdf(-1000,654,650,10)

And we got:

[tex] P(X>654)=0.3445783 [/tex]

b) For this case we assume a normal standard distribution and we want to calculate:

[tex] P(z<0.72)[/tex]

And using the following code in the Ti84 or using the normal standard table we got:

normalcdf(-1000,0.72,0,1)

[tex] P(z<0.72)=0.76424[/tex]

So this part was the wrong solution from the solution posted,

Step-by-step explanation:

Part a

We have the following distribution [tex] X \sim N(\mu =650, \sigma =10)[/tex]

And we want to calculate:

[tex] P(X>654) [/tex]

And in order to calclate this in the ti 84 we can use the following code

2nd> Vars>DISTR

And then we need to use the following code:

1-normalcdf(-1000,654,650,10)

And we got:

[tex] P(X>654)=0.3445783 [/tex]

Part b

For this case we assume a normal standard distribution and we want to calculate:

[tex] P(z<0.72)[/tex]

And using the following code in the Ti84 or using the normal standard table we got:

normalcdf(-1000,0.72,0,1)

[tex] P(z<0.72)=0.76424[/tex]

So this part was the wrong solution from the solution posted,

Answer:

A≈309.02cm² Step by step

Answer:

The distribution of scores on this final exam is left-skewed.

Step-by-step explanation:

We use the Pearson Mode Skewness to solve this question. It states that:

If the median is higher than the mean, the distribution is left-skewed.

If the median is lower than the mean, the distribution is right-skewed.

If the median is the same as the mean, the distribution is symmetric.

In this problem, we have that:

Median = 74

Mean = 70

Median higher than the mean

So the distribution of scores on this final exam is left-skewed.

Answer:

Step-by-step explanation:

median 74

mean 70

this is the answer

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.

Final answer:

Keith Bowie should deposit $667.45 into an account today to have enough money in 6 years to pay for the engine overhaul.

Explanation:

To determine the amount Keith Bowie should deposit today in an account earning 6% interest compounded annually, we can use the concept of present value. The present value of a future cash flow is calculated by dividing the future cash flow by (1 + interest rate) raised to the power of the number of years. In this case, we want to find the amount needed that would accumulate to cover the engine overhaul cost in 6 years. Let's assume the cost of the overhaul is $390 with a 10% probability, $570 with a 30% probability, $750 with a 50% probability, and $790 with a 10% probability. We can calculate the present value by multiplying each cash flow amount with its respective probability, discounting it back at the interest rate over 6 years, and adding them all up:

Present Value = (390 * 0.10) / (1 + 0.06)6 + (570 * 0.30) / (1 + 0.06)6 + (750 * 0.50) / (1 + 0.06)6 + (790 * 0.10) / (1 + 0.06)6 = $667.45

Therefore, Keith Bowie should deposit $667.45 into the account today to ensure he has enough money on hand in 6 years to pay for the engine overhaul.

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:

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:

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

The second and third function are missing and they are;

The second function; ((2^n) + n^(2))(n^(3) + 3^(n))

The third function is;

(n^(n) + n^(2n) + 5^(n))(n! + 5^(n))

Answer:

A) O((n^(3)) logn)

B) O(6^(n))

C) O(n^(n)(n!))

Step-by-step explanation:

I've attached explanation of the 3 answers.

The big-O estimate for the function f(n) = [tex]f(n) = (n log n + n^2)(n^3 + 2)[/tex] is [tex]O(n^5)[/tex] , as the product of the highest order terms [tex]n^2[/tex] and [tex]n^3[/tex] is [tex]n^5[/tex], which dominates the function's growth.

To find a big-O estimate for the function [tex]f(n) = (n log n + n^2)(n^3 + 2)[/tex], we must look for the term with the highest growth rate as n goes to infinity. This function is the product of two terms, n log [tex]n log n + n^2[/tex] and [tex]n^3 + 2[/tex]. The term with the highest growth rate in the first part is n2, and in the second part, it is n3. Therefore, the product of the two highest order terms will give us the term that grows the fastest: [tex]n^2 \times n^3 = n^5[/tex].

Thus, the given function is in big-O of [tex]n^5[/tex], which means [tex]f(n) \in O(n^5)[/tex]. The higher order term [tex]n^5[/tex]completely dominates the growth of the function, making the other terms and constants irrelevant for big-O notation. Remember that big-O notation provides an upper bound and is used to describe the worst-case scenario for the growth rate of the function.

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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:

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: 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:

[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.

Answer:

The probability that between 79% and 81%of the 500 sampled wildlife watchers actively observed mammals in 2011 is P=0.412.

Step-by-step explanation:

We know the population proportion π=0.8, according to the 2011 National Survey of Fishing, Hunting, and Wildlife-Associated Recreation.

If we take a sample from this population, and assuming the proportion is correct, it is expected that the sample's proportion to be equal to the population's proportion.

The standard deviation of the sample is equal to:

[tex]\sigma_s=\sqrt{\frac{\pi(1-\pi)}{N}}=\sqrt{\frac{0.8*0.2}{500}}=0.018[/tex]

With the mean and the standard deviaion of the sample, we can calculate the z-value for 0.79 and 0.81:

[tex]z=\frac{p-\pi}{\sigma}=\frac{0.79-0.80}{0.018} =\frac{-0.01}{0.018} = -0.55\\\\z=\frac{p-\pi}{\sigma}=\frac{0.81-0.80}{0.018} =\frac{0.01}{0.018} = 0.55[/tex]

Then, the probability that between 79% and 81%of the 500 sampled wildlife watchers actively observed mammals in 2011 is:

[tex]P(0.79<p<0.81)=P(-0.55<z<0.55)\\\\P(0.79<p<0.81)=P(z<0.55)-P(z<-0.55)\\\\P(0.79<p<0.81)=0.70884-0.29116=0.41768[/tex]

Following are the solution to the given question:

By using the approximation of normal for the proportions so, the values:  

[tex]\to P(\hat{p} < p ) = P(Z < \frac{(\hat{p} - p)}{\sqrt{\frac{p ( 1 - p)}{n}}} )[/tex]

So,

[tex]\to P(0.79 < \hat{p} < 0.81) = P( \hat{p} < 0.81) - P( \hat{p} < 0.79)[/tex]

                                  [tex]= P(Z<\frac{( 0.81 - 0.80)}{\sqrt{\frac{0.80( 1 - 0.80)}{500}}}) - P(Z < \frac{( 0.79 - 0.80)}{ \sqrt{ \frac{0.80 ( 1 - 0.80)}{500}}}) \\\\[/tex]

                                   [tex]= P(Z < 0.56) - P(Z < -0.56)\\\\= 0.7123 - 0.2877 \ \ \text{(using the Z table)}\\\\= 0.4246[/tex]

Therefore, the final answer is "0.4246".

Learn more:

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Answer:

(-1,9)

Step-by-step explanation:

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:

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:

57.93% probability that a trip will take at least 35 minutes.

Step-by-step explanation:

Problems of normally distributed samples are 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 = 36, \sigma = 4.9[/tex]

What is the probability that a trip will take at least 35 minutes

This probability is 1 subtracted by the pvalue of Z when X = 35. So

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

[tex]Z = \frac{35 - 36}{4.9}[/tex]

[tex]Z = -0.2[/tex]

[tex]Z = -0.2[/tex] has a pvalue of 0.4207

1 - 0.4207 = 0.5793

57.93% probability that a trip will take at least 35 minutes.

Answer:

Step-by-step explanation:

a. The frequency and percent frequency of complaints by industry is shown below:

Industry            Frequency    Percentage Frequency

Bank                     26                    13%

Cable                    44                    22%

Car                        42                    21%

Cell                        60                   30%

Collection              48                   13%

b. Cell has the highest complaints (60).

c. The decreasing order of complaints goes thus: Cell (60%) , Cable (22%), Car(21%).

Both Collection and Bank, together account nearly up to 27% of the total complaints.

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:

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

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:

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: